text:
@testset "4.1.1.2 (g cos)^p (a+b sin)^m" begin
    (a, b, c, d, e, f, m, p, x, ) = @variables a b c d e f m p x

    #= ::Package:: =#

    #= ::Title:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+b*sin(e+f*x))^m=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+b*sin(e+f*x))^m*when*a^2-b^2=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^p*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^7*(a + a*sin(c + d*x)), x, 3, (8*(a + a*sin(c + d*x))^5)/(5*a^4*d) - (2*(a + a*sin(c + d*x))^6)/(a^5*d) + (6*(a + a*sin(c + d*x))^7)/(7*a^6*d) - (a + a*sin(c + d*x))^8/(8*a^7*d)]
    @test_int [cos(c + d*x)^6*(a + a*sin(c + d*x)), x, 5, (5*a*x)/16 - (a*cos(c + d*x)^7)/(7*d) + (5*a*cos(c + d*x)*sin(c + d*x))/(16*d) + (5*a*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a*cos(c + d*x)^5*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x)), x, 3, (a + a*sin(c + d*x))^4/(a^3*d) - (4*(a + a*sin(c + d*x))^5)/(5*a^4*d) + (a + a*sin(c + d*x))^6/(6*a^5*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x)), x, 4, (3*a*x)/8 - (a*cos(c + d*x)^5)/(5*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x)), x, 3, (2*(a + a*sin(c + d*x))^3)/(3*a^2*d) - (a + a*sin(c + d*x))^4/(4*a^3*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x)), x, 3, (a*x)/2 - (a*cos(c + d*x)^3)/(3*d) + (a*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x)), x, 2, (a + a*sin(c + d*x))^2/(2*a*d), (a*sin(c + d*x))/d + (a*sin(c + d*x)^2)/(2*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x)), x, 2, -((a*log(1 - sin(c + d*x)))/d)]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x)), x, 3, (a*sec(c + d*x))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x)), x, 4, (a*atanh(sin(c + d*x)))/(2*d) + a^2/(2*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x)), x, 3, (a*sec(c + d*x)^3)/(3*d) + (a*tan(c + d*x))/d + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x)), x, 4, (3*a*atanh(sin(c + d*x)))/(8*d) + a^3/(8*d*(a - a*sin(c + d*x))^2) + a^2/(4*d*(a - a*sin(c + d*x))) - a^2/(8*d*(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^6*(a + a*sin(c + d*x))^2, x, 6, (45*a^2*x)/128 - (9*a^2*cos(c + d*x)^7)/(56*d) + (45*a^2*cos(c + d*x)*sin(c + d*x))/(128*d) + (15*a^2*cos(c + d*x)^3*sin(c + d*x))/(64*d) + (3*a^2*cos(c + d*x)^5*sin(c + d*x))/(16*d) - (cos(c + d*x)^7*(a^2 + a^2*sin(c + d*x)))/(8*d)]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^2, x, 3, (4*(a + a*sin(c + d*x))^5)/(5*a^3*d) - (2*(a + a*sin(c + d*x))^6)/(3*a^4*d) + (a + a*sin(c + d*x))^7/(7*a^5*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 5, (7*a^2*x)/16 - (7*a^2*cos(c + d*x)^5)/(30*d) + (7*a^2*cos(c + d*x)*sin(c + d*x))/(16*d) + (7*a^2*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (cos(c + d*x)^5*(a^2 + a^2*sin(c + d*x)))/(6*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 3, (a + a*sin(c + d*x))^4/(2*a^2*d) - (a + a*sin(c + d*x))^5/(5*a^3*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 4, (5*a^2*x)/8 - (5*a^2*cos(c + d*x)^3)/(12*d) + (5*a^2*cos(c + d*x)*sin(c + d*x))/(8*d) - (cos(c + d*x)^3*(a^2 + a^2*sin(c + d*x)))/(4*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 2, (a + a*sin(c + d*x))^3/(3*a*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^2, x, 3, (-2*a^2*log(1 - sin(c + d*x)))/d - (a^2*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^2, x, 3, (-a^2)*x + (2*a^4*cos(c + d*x))/(d*(a^2 - a^2*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^2, x, 2, a^3/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^2, x, 3, (a^4*cos(c + d*x))/(3*d*(a - a*sin(c + d*x))^2) + (a^4*cos(c + d*x))/(3*d*(a^2 - a^2*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^2, x, 4, (a^2*atanh(sin(c + d*x)))/(4*d) + a^4/(4*d*(a - a*sin(c + d*x))^2) + a^3/(4*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^6*(a + a*sin(c + d*x))^2, x, 3, (2*sec(c + d*x)^5*(a^2 + a^2*sin(c + d*x)))/(5*d) + (3*a^2*tan(c + d*x))/(5*d) + (a^2*tan(c + d*x)^3)/(5*d)]
    @test_int [sec(c + d*x)^7*(a + a*sin(c + d*x))^2, x, 4, (a^2*atanh(sin(c + d*x)))/(4*d) + a^5/(12*d*(a - a*sin(c + d*x))^3) + a^4/(8*d*(a - a*sin(c + d*x))^2) + (3*a^3)/(16*d*(a - a*sin(c + d*x))) - a^3/(16*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^8*(a + a*sin(c + d*x))^2, x, 3, (2*sec(c + d*x)^7*(a^2 + a^2*sin(c + d*x)))/(7*d) + (5*a^2*tan(c + d*x))/(7*d) + (10*a^2*tan(c + d*x)^3)/(21*d) + (a^2*tan(c + d*x)^5)/(7*d)]

    @test_int [cos(c + d*x)^6*(a + a*sin(c + d*x))^3, x, 7, (55*a^3*x)/128 - (11*a^3*cos(c + d*x)^7)/(56*d) + (55*a^3*cos(c + d*x)*sin(c + d*x))/(128*d) + (55*a^3*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (11*a^3*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (a*cos(c + d*x)^7*(a + a*sin(c + d*x))^2)/(9*d) - (11*cos(c + d*x)^7*(a^3 + a^3*sin(c + d*x)))/(72*d)]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^3, x, 3, (2*(a + a*sin(c + d*x))^6)/(3*a^3*d) - (4*(a + a*sin(c + d*x))^7)/(7*a^4*d) + (a + a*sin(c + d*x))^8/(8*a^5*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 6, (9*a^3*x)/16 - (3*a^3*cos(c + d*x)^5)/(10*d) + (9*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) + (3*a^3*cos(c + d*x)^3*sin(c + d*x))/(8*d) - (a*cos(c + d*x)^5*(a + a*sin(c + d*x))^2)/(7*d) - (3*cos(c + d*x)^5*(a^3 + a^3*sin(c + d*x)))/(14*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 3, (2*(a + a*sin(c + d*x))^5)/(5*a^2*d) - (a + a*sin(c + d*x))^6/(6*a^3*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 5, (7*a^3*x)/8 - (7*a^3*cos(c + d*x)^3)/(12*d) + (7*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) - (a*cos(c + d*x)^3*(a + a*sin(c + d*x))^2)/(5*d) - (7*cos(c + d*x)^3*(a^3 + a^3*sin(c + d*x)))/(20*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 2, (a + a*sin(c + d*x))^4/(4*a*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^3, x, 3, -((4*a^3*log(1 - sin(c + d*x)))/d) - (3*a^3*sin(c + d*x))/d - (a^3*sin(c + d*x)^2)/(2*d)]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^3, x, 4, -3*a^3*x + (3*a^3*cos(c + d*x))/d + (2*a^5*cos(c + d*x)^3)/(d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^3, x, 3, (a^3*log(1 - sin(c + d*x)))/d + (2*a^4)/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^3, x, 2, (a^6*cos(c + d*x)^3)/(3*d*(a - a*sin(c + d*x))^3)]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^3, x, 2, a^5/(2*d*(a - a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^6*(a + a*sin(c + d*x))^3, x, 4, (a^6*cos(c + d*x))/(5*d*(a - a*sin(c + d*x))^3) + (2*a^5*cos(c + d*x))/(15*d*(a - a*sin(c + d*x))^2) + (2*a^6*cos(c + d*x))/(15*d*(a^3 - a^3*sin(c + d*x)))]
    @test_int [sec(c + d*x)^7*(a + a*sin(c + d*x))^3, x, 4, (a^3*atanh(sin(c + d*x)))/(8*d) + a^6/(6*d*(a - a*sin(c + d*x))^3) + a^5/(8*d*(a - a*sin(c + d*x))^2) + a^4/(8*d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^8*(a + a*sin(c + d*x))^3, x, 4, (3*a^3*sec(c + d*x)^5)/(35*d) + (2*a*sec(c + d*x)^7*(a + a*sin(c + d*x))^2)/(7*d) + (3*a^3*tan(c + d*x))/(7*d) + (2*a^3*tan(c + d*x)^3)/(7*d) + (3*a^3*tan(c + d*x)^5)/(35*d)]

    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^8, x, 3, (4*(a + a*sin(c + d*x))^11)/(11*a^3*d) - (a + a*sin(c + d*x))^12/(3*a^4*d) + (a + a*sin(c + d*x))^13/(13*a^5*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^8, x, 11, (4199*a^8*x)/1024 - (4199*a^8*cos(c + d*x)^5)/(1920*d) + (4199*a^8*cos(c + d*x)*sin(c + d*x))/(1024*d) + (4199*a^8*cos(c + d*x)^3*sin(c + d*x))/(1536*d) - (323*a^3*cos(c + d*x)^5*(a + a*sin(c + d*x))^5)/(1320*d) - (19*a^2*cos(c + d*x)^5*(a + a*sin(c + d*x))^6)/(132*d) - (a*cos(c + d*x)^5*(a + a*sin(c + d*x))^7)/(12*d) - (4199*a^2*cos(c + d*x)^5*(a^2 + a^2*sin(c + d*x))^3)/(6336*d) - (323*cos(c + d*x)^5*(a^2 + a^2*sin(c + d*x))^4)/(792*d) - (4199*cos(c + d*x)^5*(a^4 + a^4*sin(c + d*x))^2)/(4032*d) - (4199*cos(c + d*x)^5*(a^8 + a^8*sin(c + d*x)))/(2688*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^8, x, 3, (a + a*sin(c + d*x))^10/(5*a^2*d) - (a + a*sin(c + d*x))^11/(11*a^3*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^8, x, 10, (2431*a^8*x)/256 - (2431*a^8*cos(c + d*x)^3)/(384*d) + (2431*a^8*cos(c + d*x)*sin(c + d*x))/(256*d) - (17*a^3*cos(c + d*x)^3*(a + a*sin(c + d*x))^5)/(48*d) - (17*a^2*cos(c + d*x)^3*(a + a*sin(c + d*x))^6)/(90*d) - (a*cos(c + d*x)^3*(a + a*sin(c + d*x))^7)/(10*d) - (2431*a^2*cos(c + d*x)^3*(a^2 + a^2*sin(c + d*x))^3)/(2016*d) - (221*cos(c + d*x)^3*(a^2 + a^2*sin(c + d*x))^4)/(336*d) - (2431*cos(c + d*x)^3*(a^4 + a^4*sin(c + d*x))^2)/(1120*d) - (2431*cos(c + d*x)^3*(a^8 + a^8*sin(c + d*x)))/(640*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^8, x, 2, (a + a*sin(c + d*x))^9/(9*a*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^8, x, 3, (-128*a^8*log(1 - sin(c + d*x)))/d - (64*a^8*sin(c + d*x))/d - (16*a^5*(a + a*sin(c + d*x))^3)/(3*d) - (4*a^3*(a + a*sin(c + d*x))^5)/(5*d) - (a^2*(a + a*sin(c + d*x))^6)/(3*d) - (a*(a + a*sin(c + d*x))^7)/(7*d) - (2*(a^2 + a^2*sin(c + d*x))^4)/d - (16*(a^4 + a^4*sin(c + d*x))^2)/d]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^8, x, 9, -((3003*a^8*x)/16) + (1001*a^8*cos(c + d*x)^5)/(10*d) - (3003*a^8*cos(c + d*x)*sin(c + d*x))/(16*d) - (1001*a^8*cos(c + d*x)^3*sin(c + d*x))/(8*d) + (2*a^15*cos(c + d*x)^13)/(d*(a - a*sin(c + d*x))^7) + (26*a^13*cos(c + d*x)^11)/(d*(a - a*sin(c + d*x))^5) + (286*a^14*cos(c + d*x)^9)/(3*d*(a^2 - a^2*sin(c + d*x))^3) + (143*a^16*cos(c + d*x)^7)/(2*d*(a^8 - a^8*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^8, x, 3, (192*a^8*log(1 - sin(c + d*x)))/d + (129*a^8*sin(c + d*x))/d + (36*a^8*sin(c + d*x)^2)/d + (10*a^8*sin(c + d*x)^3)/d + (2*a^8*sin(c + d*x)^4)/d + (a^8*sin(c + d*x)^5)/(5*d) + (64*a^9)/(d*(a - a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^8, x, 8, (1155*a^8*x)/8 - (385*a^8*cos(c + d*x)^3)/(4*d) + (1155*a^8*cos(c + d*x)*sin(c + d*x))/(8*d) + (2*a^15*cos(c + d*x)^11)/(3*d*(a - a*sin(c + d*x))^7) - (22*a^13*cos(c + d*x)^9)/(3*d*(a - a*sin(c + d*x))^5) - (66*a^14*cos(c + d*x)^7)/(d*(a^2 - a^2*sin(c + d*x))^3) - (231*a^16*cos(c + d*x)^5)/(4*d*(a^8 - a^8*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^8, x, 3, (-80*a^8*log(1 - sin(c + d*x)))/d - (31*a^8*sin(c + d*x))/d - (4*a^8*sin(c + d*x)^2)/d - (a^8*sin(c + d*x)^3)/(3*d) + (16*a^10)/(d*(a - a*sin(c + d*x))^2) - (80*a^9)/(d*(a - a*sin(c + d*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^6/(a + a*sin(c + d*x)), x, 4, (3*x)/(8*a) + cos(c + d*x)^5/(5*a*d) + (3*cos(c + d*x)*sin(c + d*x))/(8*a*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*a*d)]
    @test_int [cos(c + d*x)^5/(a + a*sin(c + d*x)), x, 3, -((2*(a - a*sin(c + d*x))^3)/(3*a^4*d)) + (a - a*sin(c + d*x))^4/(4*a^5*d)]
    @test_int [cos(c + d*x)^4/(a + a*sin(c + d*x)), x, 3, x/(2*a) + cos(c + d*x)^3/(3*a*d) + (cos(c + d*x)*sin(c + d*x))/(2*a*d)]
    @test_int [cos(c + d*x)^3/(a + a*sin(c + d*x)), x, 2, sin(c + d*x)/(a*d) - sin(c + d*x)^2/(2*a*d)]
    @test_int [cos(c + d*x)^2/(a + a*sin(c + d*x)), x, 2, x/a + cos(c + d*x)/(a*d)]
    @test_int [cos(c + d*x)^1/(a + a*sin(c + d*x)), x, 2, log(1 + sin(c + d*x))/(a*d)]
    @test_int [sec(c + d*x)^1/(a + a*sin(c + d*x)), x, 4, atanh(sin(c + d*x))/(2*a*d) - 1/(2*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sin(c + d*x)), x, 3, -(sec(c + d*x)/(3*d*(a + a*sin(c + d*x)))) + (2*tan(c + d*x))/(3*a*d)]
    @test_int [sec(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, (3*atanh(sin(c + d*x)))/(8*a*d) + 1/(8*d*(a - a*sin(c + d*x))) - a/(8*d*(a + a*sin(c + d*x))^2) - 1/(4*d*(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*sin(c + d*x)), x, 3, -(sec(c + d*x)^3/(5*d*(a + a*sin(c + d*x)))) + (4*tan(c + d*x))/(5*a*d) + (4*tan(c + d*x)^3)/(15*a*d)]
    @test_int [sec(c + d*x)^5/(a + a*sin(c + d*x)), x, 4, (5*atanh(sin(c + d*x)))/(16*a*d) + a/(32*d*(a - a*sin(c + d*x))^2) + 1/(8*d*(a - a*sin(c + d*x))) - a^2/(24*d*(a + a*sin(c + d*x))^3) - (3*a)/(32*d*(a + a*sin(c + d*x))^2) - 3/(16*d*(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^8/(a + a*sin(c + d*x))^2, x, 5, (7*x)/(16*a^2) + (7*cos(c + d*x)^5)/(30*a^2*d) + (7*cos(c + d*x)*sin(c + d*x))/(16*a^2*d) + (7*cos(c + d*x)^3*sin(c + d*x))/(24*a^2*d) + cos(c + d*x)^7/(6*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)^7/(a + a*sin(c + d*x))^2, x, 3, -((a - a*sin(c + d*x))^4/(2*a^6*d)) + (a - a*sin(c + d*x))^5/(5*a^7*d)]
    @test_int [cos(c + d*x)^6/(a + a*sin(c + d*x))^2, x, 4, (5*x)/(8*a^2) + (5*cos(c + d*x)^3)/(12*a^2*d) + (5*cos(c + d*x)*sin(c + d*x))/(8*a^2*d) + cos(c + d*x)^5/(4*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 2, -(a - a*sin(c + d*x))^3/(3*a^5*d)]
    @test_int [cos(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 3, (3*x)/(2*a^2) + (3*cos(c + d*x))/(2*a^2*d) + cos(c + d*x)^3/(2*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 3, (2*log(1 + sin(c + d*x)))/(a^2*d) - sin(c + d*x)/(a^2*d)]
    @test_int [cos(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 2, -(x/a^2) - (2*cos(c + d*x))/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 2, -(1/(d*(a^2 + a^2*sin(c + d*x))))]
    @test_int [sec(c + d*x)^1/(a + a*sin(c + d*x))^2, x, 4, atanh(sin(c + d*x))/(4*a^2*d) - 1/(4*d*(a + a*sin(c + d*x))^2) - 1/(4*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sin(c + d*x))^2, x, 4, -(sec(c + d*x)/(5*d*(a + a*sin(c + d*x))^2)) - sec(c + d*x)/(5*d*(a^2 + a^2*sin(c + d*x))) + (2*tan(c + d*x))/(5*a^2*d)]
    @test_int [sec(c + d*x)^3/(a + a*sin(c + d*x))^2, x, 4, atanh(sin(c + d*x))/(4*a^2*d) - a/(12*d*(a + a*sin(c + d*x))^3) - 1/(8*d*(a + a*sin(c + d*x))^2) + 1/(16*d*(a^2 - a^2*sin(c + d*x))) - 3/(16*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*sin(c + d*x))^2, x, 4, -(sec(c + d*x)^3/(7*d*(a + a*sin(c + d*x))^2)) - sec(c + d*x)^3/(7*d*(a^2 + a^2*sin(c + d*x))) + (4*tan(c + d*x))/(7*a^2*d) + (4*tan(c + d*x)^3)/(21*a^2*d)]
    @test_int [sec(c + d*x)^5/(a + a*sin(c + d*x))^2, x, 4, (15*atanh(sin(c + d*x)))/(64*a^2*d) + 1/(64*d*(a - a*sin(c + d*x))^2) - a^2/(32*d*(a + a*sin(c + d*x))^4) - a/(16*d*(a + a*sin(c + d*x))^3) - 3/(32*d*(a + a*sin(c + d*x))^2) + 5/(64*d*(a^2 - a^2*sin(c + d*x))) - 5/(32*d*(a^2 + a^2*sin(c + d*x)))]

    @test_int [cos(c + d*x)^8/(a + a*sin(c + d*x))^3, x, 5, (7*x)/(8*a^3) + (7*cos(c + d*x)^5)/(15*a^3*d) + (7*cos(c + d*x)*sin(c + d*x))/(8*a^3*d) + (7*cos(c + d*x)^3*sin(c + d*x))/(12*a^3*d) + (2*cos(c + d*x)^7)/(3*a*d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)^7/(a + a*sin(c + d*x))^3, x, 2, -((a - a*sin(c + d*x))^4/(4*a^7*d))]
    @test_int [cos(c + d*x)^6/(a + a*sin(c + d*x))^3, x, 4, (5*x)/(2*a^3) + (5*cos(c + d*x)^3)/(3*a^3*d) + (5*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + (2*cos(c + d*x)^5)/(a*d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 3, (4*log(1 + sin(c + d*x)))/(a^3*d) - (3*sin(c + d*x))/(a^3*d) + sin(c + d*x)^2/(2*a^3*d)]
    @test_int [cos(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 3, (-3*x)/a^3 - (3*cos(c + d*x))/(a^3*d) - (2*cos(c + d*x)^3)/(a*d*(a + a*sin(c + d*x))^2)]
    @test_int [cos(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 3, -(log(1 + sin(c + d*x))/(a^3*d)) - 2/(d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 1, -cos(c + d*x)^3/(3*d*(a + a*sin(c + d*x))^3)]
    @test_int [cos(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 2, -1/(2*a*d*(a + a*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^1/(a + a*sin(c + d*x))^3, x, 4, atanh(sin(c + d*x))/(8*a^3*d) - 1/(6*d*(a + a*sin(c + d*x))^3) - 1/(8*a*d*(a + a*sin(c + d*x))^2) - 1/(8*d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sin(c + d*x))^3, x, 5, -(sec(c + d*x)/(7*d*(a + a*sin(c + d*x))^3)) - (4*sec(c + d*x))/(35*a*d*(a + a*sin(c + d*x))^2) - (4*sec(c + d*x))/(35*d*(a^3 + a^3*sin(c + d*x))) + (8*tan(c + d*x))/(35*a^3*d)]
    @test_int [sec(c + d*x)^3/(a + a*sin(c + d*x))^3, x, 4, (5*atanh(sin(c + d*x)))/(32*a^3*d) - a/(16*d*(a + a*sin(c + d*x))^4) - 1/(12*d*(a + a*sin(c + d*x))^3) - 3/(32*a*d*(a + a*sin(c + d*x))^2) + 1/(32*d*(a^3 - a^3*sin(c + d*x))) - 1/(8*d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*sin(c + d*x))^3, x, 5, -(sec(c + d*x)^3/(9*d*(a + a*sin(c + d*x))^3)) - (2*sec(c + d*x)^3)/(21*a*d*(a + a*sin(c + d*x))^2) - (2*sec(c + d*x)^3)/(21*d*(a^3 + a^3*sin(c + d*x))) + (8*tan(c + d*x))/(21*a^3*d) + (8*tan(c + d*x)^3)/(63*a^3*d)]
    @test_int [sec(c + d*x)^5/(a + a*sin(c + d*x))^3, x, 4, (21*atanh(sin(c + d*x)))/(128*a^3*d) + 1/(128*a*d*(a - a*sin(c + d*x))^2) - a^2/(40*d*(a + a*sin(c + d*x))^5) - (3*a)/(64*d*(a + a*sin(c + d*x))^4) - 1/(16*d*(a + a*sin(c + d*x))^3) - 5/(64*a*d*(a + a*sin(c + d*x))^2) + 3/(64*d*(a^3 - a^3*sin(c + d*x))) - 15/(128*d*(a^3 + a^3*sin(c + d*x)))]

    @test_int [cos(c + d*x)^8/(a + a*sin(c + d*x))^8, x, 5, x/a^8 - (2*cos(c + d*x)^7)/(7*a*d*(a + a*sin(c + d*x))^7) + (2*cos(c + d*x)^5)/(5*a^3*d*(a + a*sin(c + d*x))^5) - (2*cos(c + d*x)^3)/(3*a^2*d*(a^2 + a^2*sin(c + d*x))^3) + (2*cos(c + d*x))/(d*(a^8 + a^8*sin(c + d*x)))]
    @test_int [cos(c + d*x)^7/(a + a*sin(c + d*x))^8, x, 2, -((a - a*sin(c + d*x))^4/(8*d*(a^3 + a^3*sin(c + d*x))^4))]
    @test_int [cos(c + d*x)^6/(a + a*sin(c + d*x))^8, x, 2, -(cos(c + d*x)^7/(9*d*(a + a*sin(c + d*x))^8)) - cos(c + d*x)^7/(63*a*d*(a + a*sin(c + d*x))^7)]
    @test_int [cos(c + d*x)^5/(a + a*sin(c + d*x))^8, x, 3, -4/(5*a^3*d*(a + a*sin(c + d*x))^5) - 1/(3*a^5*d*(a + a*sin(c + d*x))^3) + 1/(d*(a^2 + a^2*sin(c + d*x))^4)]
    @test_int [cos(c + d*x)^4/(a + a*sin(c + d*x))^8, x, 4, -cos(c + d*x)^5/(11*d*(a + a*sin(c + d*x))^8) - cos(c + d*x)^5/(33*a*d*(a + a*sin(c + d*x))^7) - (2*cos(c + d*x)^5)/(231*a^2*d*(a + a*sin(c + d*x))^6) - (2*cos(c + d*x)^5)/(1155*a^3*d*(a + a*sin(c + d*x))^5)]
    @test_int [cos(c + d*x)^3/(a + a*sin(c + d*x))^8, x, 3, -1/(3*a^2*d*(a + a*sin(c + d*x))^6) + 1/(5*a^3*d*(a + a*sin(c + d*x))^5)]
    @test_int [cos(c + d*x)^2/(a + a*sin(c + d*x))^8, x, 6, -cos(c + d*x)^3/(13*d*(a + a*sin(c + d*x))^8) - (5*cos(c + d*x)^3)/(143*a*d*(a + a*sin(c + d*x))^7) - (20*cos(c + d*x)^3)/(1287*a^2*d*(a + a*sin(c + d*x))^6) - (20*cos(c + d*x)^3)/(3003*a^3*d*(a + a*sin(c + d*x))^5) - (8*cos(c + d*x)^3)/(3003*d*(a^2 + a^2*sin(c + d*x))^4) - (8*cos(c + d*x)^3)/(9009*a^2*d*(a^2 + a^2*sin(c + d*x))^3)]
    @test_int [cos(c + d*x)^1/(a + a*sin(c + d*x))^8, x, 2, -1/(7*a*d*(a + a*sin(c + d*x))^7)]
    @test_int [sec(c + d*x)^1/(a + a*sin(c + d*x))^8, x, 4, atanh(sin(c + d*x))/(256*a^8*d) - 1/(16*d*(a + a*sin(c + d*x))^8) - 1/(28*a*d*(a + a*sin(c + d*x))^7) - 1/(48*a^2*d*(a + a*sin(c + d*x))^6) - 1/(80*a^3*d*(a + a*sin(c + d*x))^5) - 1/(192*a^5*d*(a + a*sin(c + d*x))^3) - 1/(128*d*(a^2 + a^2*sin(c + d*x))^4) - 1/(256*d*(a^4 + a^4*sin(c + d*x))^2) - 1/(256*d*(a^8 + a^8*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sin(c + d*x))^8, x, 10, -(sec(c + d*x)/(17*d*(a + a*sin(c + d*x))^8)) - (3*sec(c + d*x))/(85*a*d*(a + a*sin(c + d*x))^7) - (24*sec(c + d*x))/(1105*a^2*d*(a + a*sin(c + d*x))^6) - (168*sec(c + d*x))/(12155*a^3*d*(a + a*sin(c + d*x))^5) - (112*sec(c + d*x))/(12155*d*(a^2 + a^2*sin(c + d*x))^4) - (16*sec(c + d*x))/(2431*a^2*d*(a^2 + a^2*sin(c + d*x))^3) - (64*sec(c + d*x))/(12155*d*(a^4 + a^4*sin(c + d*x))^2) - (64*sec(c + d*x))/(12155*d*(a^8 + a^8*sin(c + d*x))) + (128*tan(c + d*x))/(12155*a^8*d)]
    @test_int [sec(c + d*x)^3/(a + a*sin(c + d*x))^8, x, 4, (5*atanh(sin(c + d*x)))/(512*a^8*d) - a/(36*d*(a + a*sin(c + d*x))^9) - 1/(32*d*(a + a*sin(c + d*x))^8) - 3/(112*a*d*(a + a*sin(c + d*x))^7) - 1/(48*a^2*d*(a + a*sin(c + d*x))^6) - 1/(64*a^3*d*(a + a*sin(c + d*x))^5) - 7/(768*a^5*d*(a + a*sin(c + d*x))^3) - 3/(256*d*(a^2 + a^2*sin(c + d*x))^4) - 1/(128*d*(a^4 + a^4*sin(c + d*x))^2) + 1/(1024*d*(a^8 - a^8*sin(c + d*x))) - 9/(1024*d*(a^8 + a^8*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*sin(c + d*x))^8, x, 10, -(sec(c + d*x)^3/(19*d*(a + a*sin(c + d*x))^8)) - (11*sec(c + d*x)^3)/(323*a*d*(a + a*sin(c + d*x))^7) - (22*sec(c + d*x)^3)/(969*a^2*d*(a + a*sin(c + d*x))^6) - (66*sec(c + d*x)^3)/(4199*a^3*d*(a + a*sin(c + d*x))^5) - (48*sec(c + d*x)^3)/(4199*d*(a^2 + a^2*sin(c + d*x))^4) - (112*sec(c + d*x)^3)/(12597*a^2*d*(a^2 + a^2*sin(c + d*x))^3) - (32*sec(c + d*x)^3)/(4199*d*(a^4 + a^4*sin(c + d*x))^2) - (32*sec(c + d*x)^3)/(4199*d*(a^8 + a^8*sin(c + d*x))) + (128*tan(c + d*x))/(4199*a^8*d) + (128*tan(c + d*x)^3)/(12597*a^8*d)]
    @test_int [sec(c + d*x)^5/(a + a*sin(c + d*x))^8, x, 4, (33*atanh(sin(c + d*x)))/(2048*a^8*d) - a^2/(80*d*(a + a*sin(c + d*x))^10) - a/(48*d*(a + a*sin(c + d*x))^9) - 3/(128*d*(a + a*sin(c + d*x))^8) - 5/(224*a*d*(a + a*sin(c + d*x))^7) - 5/(256*a^2*d*(a + a*sin(c + d*x))^6) - 21/(1280*a^3*d*(a + a*sin(c + d*x))^5) - 3/(256*a^5*d*(a + a*sin(c + d*x))^3) - 7/(512*d*(a^2 + a^2*sin(c + d*x))^4) + 1/(4096*d*(a^4 - a^4*sin(c + d*x))^2) - 45/(4096*d*(a^4 + a^4*sin(c + d*x))^2) + 11/(4096*d*(a^8 - a^8*sin(c + d*x))) - 55/(4096*d*(a^8 + a^8*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^p*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^7*sqrt(a + a*sin(c + d*x)), x, 3, (16*(a + a*sin(c + d*x))^(9/2))/(9*a^4*d) - (24*(a + a*sin(c + d*x))^(11/2))/(11*a^5*d) + (12*(a + a*sin(c + d*x))^(13/2))/(13*a^6*d) - (2*(a + a*sin(c + d*x))^(15/2))/(15*a^7*d)]
    @test_int [cos(c + d*x)^6*sqrt(a + a*sin(c + d*x)), x, 4, -((256*a^4*cos(c + d*x)^7)/(3003*d*(a + a*sin(c + d*x))^(7/2))) - (64*a^3*cos(c + d*x)^7)/(429*d*(a + a*sin(c + d*x))^(5/2)) - (24*a^2*cos(c + d*x)^7)/(143*d*(a + a*sin(c + d*x))^(3/2)) - (2*a*cos(c + d*x)^7)/(13*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)), x, 3, (8*(a + a*sin(c + d*x))^(7/2))/(7*a^3*d) - (8*(a + a*sin(c + d*x))^(9/2))/(9*a^4*d) + (2*(a + a*sin(c + d*x))^(11/2))/(11*a^5*d)]
    @test_int [cos(c + d*x)^4*sqrt(a + a*sin(c + d*x)), x, 3, (-64*a^3*cos(c + d*x)^5)/(315*d*(a + a*sin(c + d*x))^(5/2)) - (16*a^2*cos(c + d*x)^5)/(63*d*(a + a*sin(c + d*x))^(3/2)) - (2*a*cos(c + d*x)^5)/(9*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 3, (4*(a + a*sin(c + d*x))^(5/2))/(5*a^2*d) - (2*(a + a*sin(c + d*x))^(7/2))/(7*a^3*d)]
    @test_int [cos(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 2, (-8*a^2*cos(c + d*x)^3)/(15*d*(a + a*sin(c + d*x))^(3/2)) - (2*a*cos(c + d*x)^3)/(5*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 2, (2*(a + a*sin(c + d*x))^(3/2))/(3*a*d)]
    @test_int [sec(c + d*x)^1*sqrt(a + a*sin(c + d*x)), x, 3, (sqrt(2)*sqrt(a)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/d]
    @test_int [sec(c + d*x)^2*sqrt(a + a*sin(c + d*x)), x, 3, -((sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(2)*d)) + (sec(c + d*x)*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^3*sqrt(a + a*sin(c + d*x)), x, 5, (3*sqrt(a)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(4*sqrt(2)*d) - (3*a)/(4*d*sqrt(a + a*sin(c + d*x))) + (sec(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(2*d)]
    @test_int [sec(c + d*x)^4*sqrt(a + a*sin(c + d*x)), x, 5, -((5*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(8*sqrt(2)*d)) - (5*a^2*cos(c + d*x))/(8*d*(a + a*sin(c + d*x))^(3/2)) + (5*a*sec(c + d*x))/(6*d*sqrt(a + a*sin(c + d*x))) + (sec(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(3*d)]
    @test_int [sec(c + d*x)^5*sqrt(a + a*sin(c + d*x)), x, 7, (35*sqrt(a)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(64*sqrt(2)*d) - (35*a^2)/(96*d*(a + a*sin(c + d*x))^(3/2)) - (35*a)/(64*d*sqrt(a + a*sin(c + d*x))) + (7*a*sec(c + d*x)^2)/(16*d*sqrt(a + a*sin(c + d*x))) + (sec(c + d*x)^4*sqrt(a + a*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^6*sqrt(a + a*sin(c + d*x)), x, 7, -((63*sqrt(a)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(128*sqrt(2)*d)) - (63*a^2*cos(c + d*x))/(128*d*(a + a*sin(c + d*x))^(3/2)) - (21*a^2*sec(c + d*x))/(80*d*(a + a*sin(c + d*x))^(3/2)) + (21*a*sec(c + d*x))/(32*d*sqrt(a + a*sin(c + d*x))) + (3*a*sec(c + d*x)^3)/(10*d*sqrt(a + a*sin(c + d*x))) + (sec(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(5*d)]

    @test_int [cos(c + d*x)^7*(a + a*sin(c + d*x))^(3/2), x, 3, (16*(a + a*sin(c + d*x))^(11/2))/(11*a^4*d) - (24*(a + a*sin(c + d*x))^(13/2))/(13*a^5*d) + (4*(a + a*sin(c + d*x))^(15/2))/(5*a^6*d) - (2*(a + a*sin(c + d*x))^(17/2))/(17*a^7*d)]
    @test_int [cos(c + d*x)^6*(a + a*sin(c + d*x))^(3/2), x, 5, -((4096*a^5*cos(c + d*x)^7)/(45045*d*(a + a*sin(c + d*x))^(7/2))) - (1024*a^4*cos(c + d*x)^7)/(6435*d*(a + a*sin(c + d*x))^(5/2)) - (128*a^3*cos(c + d*x)^7)/(715*d*(a + a*sin(c + d*x))^(3/2)) - (32*a^2*cos(c + d*x)^7)/(195*d*sqrt(a + a*sin(c + d*x))) - (2*a*cos(c + d*x)^7*sqrt(a + a*sin(c + d*x)))/(15*d)]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^(3/2), x, 3, (8*(a + a*sin(c + d*x))^(9/2))/(9*a^3*d) - (8*(a + a*sin(c + d*x))^(11/2))/(11*a^4*d) + (2*(a + a*sin(c + d*x))^(13/2))/(13*a^5*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^(3/2), x, 4, (-256*a^4*cos(c + d*x)^5)/(1155*d*(a + a*sin(c + d*x))^(5/2)) - (64*a^3*cos(c + d*x)^5)/(231*d*(a + a*sin(c + d*x))^(3/2)) - (8*a^2*cos(c + d*x)^5)/(33*d*sqrt(a + a*sin(c + d*x))) - (2*a*cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(11*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 3, (4*(a + a*sin(c + d*x))^(7/2))/(7*a^2*d) - (2*(a + a*sin(c + d*x))^(9/2))/(9*a^3*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 3, (-64*a^3*cos(c + d*x)^3)/(105*d*(a + a*sin(c + d*x))^(3/2)) - (16*a^2*cos(c + d*x)^3)/(35*d*sqrt(a + a*sin(c + d*x))) - (2*a*cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(7*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 2, (2*(a + a*sin(c + d*x))^(5/2))/(5*a*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^(3/2), x, 4, (2*sqrt(2)*a^(3/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/d - (2*a*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^(3/2), x, 1, (2*a*sec(c + d*x)*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^(3/2), x, 4, (a^(3/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(2*sqrt(2)*d) + (sec(c + d*x)^2*(a + a*sin(c + d*x))^(3/2))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^(3/2), x, 4, -((a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(2*sqrt(2)*d)) + (a*sec(c + d*x)*sqrt(a + a*sin(c + d*x)))/(2*d) + (sec(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(3*d)]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^(3/2), x, 6, (15*a^(3/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(32*sqrt(2)*d) - (15*a^2)/(32*d*sqrt(a + a*sin(c + d*x))) + (5*a*sec(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(16*d) + (sec(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/(4*d)]
    @test_int [sec(c + d*x)^6*(a + a*sin(c + d*x))^(3/2), x, 6, -((7*a^(3/2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*d)) - (7*a^3*cos(c + d*x))/(16*d*(a + a*sin(c + d*x))^(3/2)) + (7*a^2*sec(c + d*x))/(12*d*sqrt(a + a*sin(c + d*x))) + (7*a*sec(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(30*d) + (sec(c + d*x)^5*(a + a*sin(c + d*x))^(3/2))/(5*d)]

    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^(5/2), x, 3, (8*(a + a*sin(c + d*x))^(11/2))/(11*a^3*d) - (8*(a + a*sin(c + d*x))^(13/2))/(13*a^4*d) + (2*(a + a*sin(c + d*x))^(15/2))/(15*a^5*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^(5/2), x, 5, (-4096*a^5*cos(c + d*x)^5)/(15015*d*(a + a*sin(c + d*x))^(5/2)) - (1024*a^4*cos(c + d*x)^5)/(3003*d*(a + a*sin(c + d*x))^(3/2)) - (128*a^3*cos(c + d*x)^5)/(429*d*sqrt(a + a*sin(c + d*x))) - (32*a^2*cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(143*d) - (2*a*cos(c + d*x)^5*(a + a*sin(c + d*x))^(3/2))/(13*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^(5/2), x, 3, (4*(a + a*sin(c + d*x))^(9/2))/(9*a^2*d) - (2*(a + a*sin(c + d*x))^(11/2))/(11*a^3*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^(5/2), x, 4, (-256*a^4*cos(c + d*x)^3)/(315*d*(a + a*sin(c + d*x))^(3/2)) - (64*a^3*cos(c + d*x)^3)/(105*d*sqrt(a + a*sin(c + d*x))) - (8*a^2*cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(21*d) - (2*a*cos(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(9*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^(5/2), x, 2, (2*(a + a*sin(c + d*x))^(7/2))/(7*a*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^(5/2), x, 5, (4*sqrt(2)*a^(5/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/d - (4*a^2*sqrt(a + a*sin(c + d*x)))/d - (2*a*(a + a*sin(c + d*x))^(3/2))/(3*d)]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^(5/2), x, 2, (8*a^2*sec(c + d*x)*sqrt(a + a*sin(c + d*x)))/d - (2*a*sec(c + d*x)*(a + a*sin(c + d*x))^(3/2))/d]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^(5/2), x, 4, -((a^(5/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(sqrt(2)*d)) + (a*sec(c + d*x)^2*(a + a*sin(c + d*x))^(3/2))/d]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^(5/2), x, 1, (2*a*sec(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(3*d)]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^(5/2), x, 5, (3*a^(5/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(16*sqrt(2)*d) + (3*a*sec(c + d*x)^2*(a + a*sin(c + d*x))^(3/2))/(16*d) + (sec(c + d*x)^4*(a + a*sin(c + d*x))^(5/2))/(4*d)]
    @test_int [sec(c + d*x)^6*(a + a*sin(c + d*x))^(5/2), x, 5, -((a^(5/2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(4*sqrt(2)*d)) + (a^2*sec(c + d*x)*sqrt(a + a*sin(c + d*x)))/(4*d) + (a*sec(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(6*d) + (sec(c + d*x)^5*(a + a*sin(c + d*x))^(5/2))/(5*d)]
    @test_int [sec(c + d*x)^7*(a + a*sin(c + d*x))^(5/2), x, 7, (35*a^(5/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(128*sqrt(2)*d) - (35*a^3)/(128*d*sqrt(a + a*sin(c + d*x))) + (35*a^2*sec(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(192*d) + (7*a*sec(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/(48*d) + (sec(c + d*x)^6*(a + a*sin(c + d*x))^(5/2))/(6*d)]

    @test_int [cos(c + d*x)^7*(a + a*sin(c + d*x))^(7/2), x, 3, (16*(a + a*sin(c + d*x))^(15/2))/(15*a^4*d) - (24*(a + a*sin(c + d*x))^(17/2))/(17*a^5*d) + (12*(a + a*sin(c + d*x))^(19/2))/(19*a^6*d) - (2*(a + a*sin(c + d*x))^(21/2))/(21*a^7*d)]
    @test_int [cos(c + d*x)^6*(a + a*sin(c + d*x))^(7/2), x, 7, -((131072*a^7*cos(c + d*x)^7)/(969969*d*(a + a*sin(c + d*x))^(7/2))) - (32768*a^6*cos(c + d*x)^7)/(138567*d*(a + a*sin(c + d*x))^(5/2)) - (12288*a^5*cos(c + d*x)^7)/(46189*d*(a + a*sin(c + d*x))^(3/2)) - (1024*a^4*cos(c + d*x)^7)/(4199*d*sqrt(a + a*sin(c + d*x))) - (64*a^3*cos(c + d*x)^7*sqrt(a + a*sin(c + d*x)))/(323*d) - (48*a^2*cos(c + d*x)^7*(a + a*sin(c + d*x))^(3/2))/(323*d) - (2*a*cos(c + d*x)^7*(a + a*sin(c + d*x))^(5/2))/(19*d)]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^(7/2), x, 3, (8*(a + a*sin(c + d*x))^(13/2))/(13*a^3*d) - (8*(a + a*sin(c + d*x))^(15/2))/(15*a^4*d) + (2*(a + a*sin(c + d*x))^(17/2))/(17*a^5*d)]
    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^(7/2), x, 6, -((16384*a^6*cos(c + d*x)^5)/(45045*d*(a + a*sin(c + d*x))^(5/2))) - (4096*a^5*cos(c + d*x)^5)/(9009*d*(a + a*sin(c + d*x))^(3/2)) - (512*a^4*cos(c + d*x)^5)/(1287*d*sqrt(a + a*sin(c + d*x))) - (128*a^3*cos(c + d*x)^5*sqrt(a + a*sin(c + d*x)))/(429*d) - (8*a^2*cos(c + d*x)^5*(a + a*sin(c + d*x))^(3/2))/(39*d) - (2*a*cos(c + d*x)^5*(a + a*sin(c + d*x))^(5/2))/(15*d)]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^(7/2), x, 3, (4*(a + a*sin(c + d*x))^(11/2))/(11*a^2*d) - (2*(a + a*sin(c + d*x))^(13/2))/(13*a^3*d)]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^(7/2), x, 5, -((4096*a^5*cos(c + d*x)^3)/(3465*d*(a + a*sin(c + d*x))^(3/2))) - (1024*a^4*cos(c + d*x)^3)/(1155*d*sqrt(a + a*sin(c + d*x))) - (128*a^3*cos(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(231*d) - (32*a^2*cos(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(99*d) - (2*a*cos(c + d*x)^3*(a + a*sin(c + d*x))^(5/2))/(11*d)]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^(7/2), x, 2, (2*(a + a*sin(c + d*x))^(9/2))/(9*a*d)]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^(7/2), x, 6, (8*sqrt(2)*a^(7/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/d - (8*a^3*sqrt(a + a*sin(c + d*x)))/d - (4*a^2*(a + a*sin(c + d*x))^(3/2))/(3*d) - (2*a*(a + a*sin(c + d*x))^(5/2))/(5*d)]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^(7/2), x, 3, (64*a^3*sec(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d) - (16*a^2*sec(c + d*x)*(a + a*sin(c + d*x))^(3/2))/(3*d) - (2*a*sec(c + d*x)*(a + a*sin(c + d*x))^(5/2))/(3*d)]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^(7/2), x, 5, -((3*sqrt(2)*a^(7/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/d) + (3*a^3*sqrt(a + a*sin(c + d*x)))/d + (a*sec(c + d*x)^2*(a + a*sin(c + d*x))^(5/2))/d]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^(7/2), x, 2, -((8*a^2*sec(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(3*d)) + (2*a*sec(c + d*x)^3*(a + a*sin(c + d*x))^(5/2))/d]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^(7/2), x, 5, -((a^(7/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(8*sqrt(2)*d)) - (a^2*sec(c + d*x)^2*(a + a*sin(c + d*x))^(3/2))/(8*d) + (a*sec(c + d*x)^4*(a + a*sin(c + d*x))^(5/2))/(2*d)]
    @test_int [sec(c + d*x)^6*(a + a*sin(c + d*x))^(7/2), x, 1, (2*a*sec(c + d*x)^5*(a + a*sin(c + d*x))^(5/2))/(5*d)]
    @test_int [sec(c + d*x)^7*(a + a*sin(c + d*x))^(7/2), x, 6, (5*a^(7/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(64*sqrt(2)*d) + (5*a^2*sec(c + d*x)^2*(a + a*sin(c + d*x))^(3/2))/(64*d) + (5*a*sec(c + d*x)^4*(a + a*sin(c + d*x))^(5/2))/(48*d) + (sec(c + d*x)^6*(a + a*sin(c + d*x))^(7/2))/(6*d)]
    @test_int [sec(c + d*x)^8*(a + a*sin(c + d*x))^(7/2), x, 6, -((a^(7/2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(8*sqrt(2)*d)) + (a^3*sec(c + d*x)*sqrt(a + a*sin(c + d*x)))/(8*d) + (a^2*sec(c + d*x)^3*(a + a*sin(c + d*x))^(3/2))/(12*d) + (a*sec(c + d*x)^5*(a + a*sin(c + d*x))^(5/2))/(10*d) + (sec(c + d*x)^7*(a + a*sin(c + d*x))^(7/2))/(7*d)]
    @test_int [sec(c + d*x)^9*(a + a*sin(c + d*x))^(7/2), x, 8, (315*a^(7/2)*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(2048*sqrt(2)*d) - (315*a^4)/(2048*d*sqrt(a + a*sin(c + d*x))) + (105*a^3*sec(c + d*x)^2*sqrt(a + a*sin(c + d*x)))/(1024*d) + (21*a^2*sec(c + d*x)^4*(a + a*sin(c + d*x))^(3/2))/(256*d) + (3*a*sec(c + d*x)^6*(a + a*sin(c + d*x))^(5/2))/(32*d) + (sec(c + d*x)^8*(a + a*sin(c + d*x))^(7/2))/(8*d)]
    @test_int [sec(c + d*x)^10*(a + a*sin(c + d*x))^(7/2), x, 8, -((11*a^(7/2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(64*sqrt(2)*d)) - (11*a^5*cos(c + d*x))/(64*d*(a + a*sin(c + d*x))^(3/2)) + (11*a^4*sec(c + d*x))/(48*d*sqrt(a + a*sin(c + d*x))) + (11*a^3*sec(c + d*x)^3*sqrt(a + a*sin(c + d*x)))/(120*d) + (11*a^2*sec(c + d*x)^5*(a + a*sin(c + d*x))^(3/2))/(140*d) + (11*a*sec(c + d*x)^7*(a + a*sin(c + d*x))^(5/2))/(126*d) + (sec(c + d*x)^9*(a + a*sin(c + d*x))^(7/2))/(9*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^7/sqrt(a + a*sin(c + d*x)), x, 3, (16*(a + a*sin(c + d*x))^(7/2))/(7*a^4*d) - (8*(a + a*sin(c + d*x))^(9/2))/(3*a^5*d) + (12*(a + a*sin(c + d*x))^(11/2))/(11*a^6*d) - (2*(a + a*sin(c + d*x))^(13/2))/(13*a^7*d)]
    @test_int [cos(c + d*x)^6/sqrt(a + a*sin(c + d*x)), x, 3, -((64*a^3*cos(c + d*x)^7)/(693*d*(a + a*sin(c + d*x))^(7/2))) - (16*a^2*cos(c + d*x)^7)/(99*d*(a + a*sin(c + d*x))^(5/2)) - (2*a*cos(c + d*x)^7)/(11*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^5/sqrt(a + a*sin(c + d*x)), x, 3, (8*(a + a*sin(c + d*x))^(5/2))/(5*a^3*d) - (8*(a + a*sin(c + d*x))^(7/2))/(7*a^4*d) + (2*(a + a*sin(c + d*x))^(9/2))/(9*a^5*d)]
    @test_int [cos(c + d*x)^4/sqrt(a + a*sin(c + d*x)), x, 2, (-8*a^2*cos(c + d*x)^5)/(35*d*(a + a*sin(c + d*x))^(5/2)) - (2*a*cos(c + d*x)^5)/(7*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 3, (4*(a + a*sin(c + d*x))^(3/2))/(3*a^2*d) - (2*(a + a*sin(c + d*x))^(5/2))/(5*a^3*d)]
    @test_int [cos(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 1, (-2*a*cos(c + d*x)^3)/(3*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 2, (2*sqrt(a + a*sin(c + d*x)))/(a*d)]
    @test_int [sec(c + d*x)^1/sqrt(a + a*sin(c + d*x)), x, 4, atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a)))/(sqrt(2)*sqrt(a)*d) - 1/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/sqrt(a + a*sin(c + d*x)), x, 4, -((3*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(4*sqrt(2)*sqrt(a)*d)) - (3*a*cos(c + d*x))/(4*d*(a + a*sin(c + d*x))^(3/2)) + sec(c + d*x)/(d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3/sqrt(a + a*sin(c + d*x)), x, 6, (5*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(8*sqrt(2)*sqrt(a)*d) - (5*a)/(12*d*(a + a*sin(c + d*x))^(3/2)) - 5/(8*d*sqrt(a + a*sin(c + d*x))) + sec(c + d*x)^2/(2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4/sqrt(a + a*sin(c + d*x)), x, 6, -((35*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(64*sqrt(2)*sqrt(a)*d)) - (35*a*cos(c + d*x))/(64*d*(a + a*sin(c + d*x))^(3/2)) - (7*a*sec(c + d*x))/(24*d*(a + a*sin(c + d*x))^(3/2)) + (35*sec(c + d*x))/(48*d*sqrt(a + a*sin(c + d*x))) + sec(c + d*x)^3/(3*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5/sqrt(a + a*sin(c + d*x)), x, 8, (63*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(128*sqrt(2)*sqrt(a)*d) - (21*a)/(64*d*(a + a*sin(c + d*x))^(3/2)) - (9*a*sec(c + d*x)^2)/(40*d*(a + a*sin(c + d*x))^(3/2)) - 63/(128*d*sqrt(a + a*sin(c + d*x))) + (63*sec(c + d*x)^2)/(160*d*sqrt(a + a*sin(c + d*x))) + sec(c + d*x)^4/(4*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^6/sqrt(a + a*sin(c + d*x)), x, 8, -((231*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(512*sqrt(2)*sqrt(a)*d)) - (231*a*cos(c + d*x))/(512*d*(a + a*sin(c + d*x))^(3/2)) - (77*a*sec(c + d*x))/(320*d*(a + a*sin(c + d*x))^(3/2)) - (11*a*sec(c + d*x)^3)/(60*d*(a + a*sin(c + d*x))^(3/2)) + (77*sec(c + d*x))/(128*d*sqrt(a + a*sin(c + d*x))) + (11*sec(c + d*x)^3)/(40*d*sqrt(a + a*sin(c + d*x))) + sec(c + d*x)^5/(5*d*sqrt(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^7/(a + a*sin(c + d*x))^(3/2), x, 3, (16*(a + a*sin(c + d*x))^(5/2))/(5*a^4*d) - (24*(a + a*sin(c + d*x))^(7/2))/(7*a^5*d) + (4*(a + a*sin(c + d*x))^(9/2))/(3*a^6*d) - (2*(a + a*sin(c + d*x))^(11/2))/(11*a^7*d)]
    @test_int [cos(c + d*x)^6/(a + a*sin(c + d*x))^(3/2), x, 2, -((8*a^2*cos(c + d*x)^7)/(63*d*(a + a*sin(c + d*x))^(7/2))) - (2*a*cos(c + d*x)^7)/(9*d*(a + a*sin(c + d*x))^(5/2))]
    @test_int [cos(c + d*x)^5/(a + a*sin(c + d*x))^(3/2), x, 3, (8*(a + a*sin(c + d*x))^(3/2))/(3*a^3*d) - (8*(a + a*sin(c + d*x))^(5/2))/(5*a^4*d) + (2*(a + a*sin(c + d*x))^(7/2))/(7*a^5*d)]
    @test_int [cos(c + d*x)^4/(a + a*sin(c + d*x))^(3/2), x, 1, (-2*a*cos(c + d*x)^5)/(5*d*(a + a*sin(c + d*x))^(5/2))]
    @test_int [cos(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 3, (4*sqrt(a + a*sin(c + d*x)))/(a^2*d) - (2*(a + a*sin(c + d*x))^(3/2))/(3*a^3*d)]
    @test_int [cos(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 3, (-2*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(3/2)*d) + (2*cos(c + d*x))/(a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 2, -2/(a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*sin(c + d*x))^(3/2), x, 5, atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a)))/(2*sqrt(2)*a^(3/2)*d) - 1/(3*d*(a + a*sin(c + d*x))^(3/2)) - 1/(2*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sin(c + d*x))^(3/2), x, 5, -((15*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(32*sqrt(2)*a^(3/2)*d)) - (15*cos(c + d*x))/(32*d*(a + a*sin(c + d*x))^(3/2)) - sec(c + d*x)/(4*d*(a + a*sin(c + d*x))^(3/2)) + (5*sec(c + d*x))/(8*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*sin(c + d*x))^(3/2), x, 7, (7*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(16*sqrt(2)*a^(3/2)*d) - 7/(24*d*(a + a*sin(c + d*x))^(3/2)) - sec(c + d*x)^2/(5*d*(a + a*sin(c + d*x))^(3/2)) - 7/(16*a*d*sqrt(a + a*sin(c + d*x))) + (7*sec(c + d*x)^2)/(20*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*sin(c + d*x))^(3/2), x, 7, -((105*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(256*sqrt(2)*a^(3/2)*d)) - (105*cos(c + d*x))/(256*d*(a + a*sin(c + d*x))^(3/2)) - (7*sec(c + d*x))/(32*d*(a + a*sin(c + d*x))^(3/2)) - sec(c + d*x)^3/(6*d*(a + a*sin(c + d*x))^(3/2)) + (35*sec(c + d*x))/(64*a*d*sqrt(a + a*sin(c + d*x))) + sec(c + d*x)^3/(4*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5/(a + a*sin(c + d*x))^(3/2), x, 9, (99*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(256*sqrt(2)*a^(3/2)*d) - 33/(128*d*(a + a*sin(c + d*x))^(3/2)) - (99*sec(c + d*x)^2)/(560*d*(a + a*sin(c + d*x))^(3/2)) - sec(c + d*x)^4/(7*d*(a + a*sin(c + d*x))^(3/2)) - 99/(256*a*d*sqrt(a + a*sin(c + d*x))) + (99*sec(c + d*x)^2)/(320*a*d*sqrt(a + a*sin(c + d*x))) + (11*sec(c + d*x)^4)/(56*a*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^6/(a + a*sin(c + d*x))^(3/2), x, 9, -((3003*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(8192*sqrt(2)*a^(3/2)*d)) - (3003*cos(c + d*x))/(8192*d*(a + a*sin(c + d*x))^(3/2)) - (1001*sec(c + d*x))/(5120*d*(a + a*sin(c + d*x))^(3/2)) - (143*sec(c + d*x)^3)/(960*d*(a + a*sin(c + d*x))^(3/2)) - sec(c + d*x)^5/(8*d*(a + a*sin(c + d*x))^(3/2)) + (1001*sec(c + d*x))/(2048*a*d*sqrt(a + a*sin(c + d*x))) + (143*sec(c + d*x)^3)/(640*a*d*sqrt(a + a*sin(c + d*x))) + (13*sec(c + d*x)^5)/(80*a*d*sqrt(a + a*sin(c + d*x)))]

    @test_int [cos(c + d*x)^10/(a + a*sin(c + d*x))^(5/2), x, 3, -((64*a^3*cos(c + d*x)^11)/(2145*d*(a + a*sin(c + d*x))^(11/2))) - (16*a^2*cos(c + d*x)^11)/(195*d*(a + a*sin(c + d*x))^(9/2)) - (2*a*cos(c + d*x)^11)/(15*d*(a + a*sin(c + d*x))^(7/2))]
    @test_int [cos(c + d*x)^9/(a + a*sin(c + d*x))^(5/2), x, 3, (32*(a + a*sin(c + d*x))^(5/2))/(5*a^5*d) - (64*(a + a*sin(c + d*x))^(7/2))/(7*a^6*d) + (16*(a + a*sin(c + d*x))^(9/2))/(3*a^7*d) - (16*(a + a*sin(c + d*x))^(11/2))/(11*a^8*d) + (2*(a + a*sin(c + d*x))^(13/2))/(13*a^9*d)]
    @test_int [cos(c + d*x)^8/(a + a*sin(c + d*x))^(5/2), x, 2, -((8*a^2*cos(c + d*x)^9)/(99*d*(a + a*sin(c + d*x))^(9/2))) - (2*a*cos(c + d*x)^9)/(11*d*(a + a*sin(c + d*x))^(7/2))]
    @test_int [cos(c + d*x)^7/(a + a*sin(c + d*x))^(5/2), x, 3, (16*(a + a*sin(c + d*x))^(3/2))/(3*a^4*d) - (24*(a + a*sin(c + d*x))^(5/2))/(5*a^5*d) + (12*(a + a*sin(c + d*x))^(7/2))/(7*a^6*d) - (2*(a + a*sin(c + d*x))^(9/2))/(9*a^7*d)]
    @test_int [cos(c + d*x)^6/(a + a*sin(c + d*x))^(5/2), x, 1, -((2*a*cos(c + d*x)^7)/(7*d*(a + a*sin(c + d*x))^(7/2)))]
    @test_int [cos(c + d*x)^5/(a + a*sin(c + d*x))^(5/2), x, 3, (8*sqrt(a + a*sin(c + d*x)))/(a^3*d) - (8*(a + a*sin(c + d*x))^(3/2))/(3*a^4*d) + (2*(a + a*sin(c + d*x))^(5/2))/(5*a^5*d)]
    @test_int [cos(c + d*x)^4/(a + a*sin(c + d*x))^(5/2), x, 4, (-4*sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(a^(5/2)*d) + (2*cos(c + d*x)^3)/(3*a*d*(a + a*sin(c + d*x))^(3/2)) + (4*cos(c + d*x))/(a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + a*sin(c + d*x))^(5/2), x, 3, -4/(a^2*d*sqrt(a + a*sin(c + d*x))) - (2*sqrt(a + a*sin(c + d*x)))/(a^3*d)]
    @test_int [cos(c + d*x)^2/(a + a*sin(c + d*x))^(5/2), x, 3, atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x))))/(sqrt(2)*a^(5/2)*d) - cos(c + d*x)/(a*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^1/(a + a*sin(c + d*x))^(5/2), x, 2, -2/(3*a*d*(a + a*sin(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^1/(a + a*sin(c + d*x))^(5/2), x, 6, atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a)))/(4*sqrt(2)*a^(5/2)*d) - 1/(5*d*(a + a*sin(c + d*x))^(5/2)) - 1/(6*a*d*(a + a*sin(c + d*x))^(3/2)) - 1/(4*a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*sin(c + d*x))^(5/2), x, 6, -((35*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(128*sqrt(2)*a^(5/2)*d)) - sec(c + d*x)/(6*d*(a + a*sin(c + d*x))^(5/2)) - (35*cos(c + d*x))/(128*a*d*(a + a*sin(c + d*x))^(3/2)) - (7*sec(c + d*x))/(48*a*d*(a + a*sin(c + d*x))^(3/2)) + (35*sec(c + d*x))/(96*a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*sin(c + d*x))^(5/2), x, 8, (9*atanh(sqrt(a + a*sin(c + d*x))/(sqrt(2)*sqrt(a))))/(32*sqrt(2)*a^(5/2)*d) - sec(c + d*x)^2/(7*d*(a + a*sin(c + d*x))^(5/2)) - 3/(16*a*d*(a + a*sin(c + d*x))^(3/2)) - (9*sec(c + d*x)^2)/(70*a*d*(a + a*sin(c + d*x))^(3/2)) - 9/(32*a^2*d*sqrt(a + a*sin(c + d*x))) + (9*sec(c + d*x)^2)/(40*a^2*d*sqrt(a + a*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*sin(c + d*x))^(5/2), x, 8, -((1155*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(4096*sqrt(2)*a^(5/2)*d)) - sec(c + d*x)^3/(8*d*(a + a*sin(c + d*x))^(5/2)) - (1155*cos(c + d*x))/(4096*a*d*(a + a*sin(c + d*x))^(3/2)) - (77*sec(c + d*x))/(512*a*d*(a + a*sin(c + d*x))^(3/2)) - (11*sec(c + d*x)^3)/(96*a*d*(a + a*sin(c + d*x))^(3/2)) + (385*sec(c + d*x))/(1024*a^2*d*sqrt(a + a*sin(c + d*x))) + (11*sec(c + d*x)^3)/(64*a^2*d*sqrt(a + a*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^(p/2)*(a+a*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x)), x, 5, (-2*a*(e*cos(c + d*x))^(9/2))/(9*d*e) + (10*a*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*d*sqrt(e*cos(c + d*x))) + (10*a*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*e*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x)), x, 4, (-2*a*(e*cos(c + d*x))^(7/2))/(7*d*e) + (6*a*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*sqrt(cos(c + d*x))) + (2*a*e*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x)), x, 4, (-2*a*(e*cos(c + d*x))^(5/2))/(5*d*e) + (2*a*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*sqrt(e*cos(c + d*x))) + (2*a*e*sqrt(e*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x)), x, 3, (-2*a*(e*cos(c + d*x))^(3/2))/(3*d*e) + (2*a*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))/sqrt(e*cos(c + d*x)), x, 3, (-2*a*sqrt(e*cos(c + d*x)))/(d*e) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(d*sqrt(e*cos(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))/(e*cos(c + d*x))^(3/2), x, 4, (2*a)/(d*e*sqrt(e*cos(c + d*x))) - (2*a*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(d*e^2*sqrt(cos(c + d*x))) + (2*a*sin(c + d*x))/(d*e*sqrt(e*cos(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))/(e*cos(c + d*x))^(5/2), x, 4, (2*a)/(3*d*e*(e*cos(c + d*x))^(3/2)) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*e^2*sqrt(e*cos(c + d*x))) + (2*a*sin(c + d*x))/(3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + a*sin(c + d*x))/(e*cos(c + d*x))^(7/2), x, 5, (2*a)/(5*d*e*(e*cos(c + d*x))^(5/2)) - (6*a*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*e^4*sqrt(cos(c + d*x))) + (2*a*sin(c + d*x))/(5*d*e*(e*cos(c + d*x))^(5/2)) + (6*a*sin(c + d*x))/(5*d*e^3*sqrt(e*cos(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))^2, x, 6, (-26*a^2*(e*cos(c + d*x))^(9/2))/(99*d*e) + (130*a^2*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(231*d*sqrt(e*cos(c + d*x))) + (130*a^2*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(231*d) + (26*a^2*e*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(77*d) - (2*(e*cos(c + d*x))^(9/2)*(a^2 + a^2*sin(c + d*x)))/(11*d*e)]
    @test_int [(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^2, x, 5, (-22*a^2*(e*cos(c + d*x))^(7/2))/(63*d*e) + (22*a^2*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(15*d*sqrt(cos(c + d*x))) + (22*a^2*e*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(45*d) - (2*(e*cos(c + d*x))^(7/2)*(a^2 + a^2*sin(c + d*x)))/(9*d*e)]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^2, x, 5, (-18*a^2*(e*cos(c + d*x))^(5/2))/(35*d*e) + (6*a^2*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(7*d*sqrt(e*cos(c + d*x))) + (6*a^2*e*sqrt(e*cos(c + d*x))*sin(c + d*x))/(7*d) - (2*(e*cos(c + d*x))^(5/2)*(a^2 + a^2*sin(c + d*x)))/(7*d*e)]
    @test_int [sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^2, x, 4, (-14*a^2*(e*cos(c + d*x))^(3/2))/(15*d*e) + (14*a^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*sqrt(cos(c + d*x))) - (2*(e*cos(c + d*x))^(3/2)*(a^2 + a^2*sin(c + d*x)))/(5*d*e)]
    @test_int [(a + a*sin(c + d*x))^2/sqrt(e*cos(c + d*x)), x, 4, (-10*a^2*sqrt(e*cos(c + d*x)))/(3*d*e) + (10*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*sqrt(e*cos(c + d*x))) - (2*sqrt(e*cos(c + d*x))*(a^2 + a^2*sin(c + d*x)))/(3*d*e)]
    @test_int [(a + a*sin(c + d*x))^2/(e*cos(c + d*x))^(3/2), x, 4, -((6*a^2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*e^2*sqrt(cos(c + d*x)))) + (4*a^4*(e*cos(c + d*x))^(3/2))/(d*e^3*(a^2 - a^2*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^2/(e*cos(c + d*x))^(5/2), x, 4, -((2*a^2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*e^2*sqrt(e*cos(c + d*x)))) + (4*a^4*sqrt(e*cos(c + d*x)))/(3*d*e^3*(a^2 - a^2*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^2/(e*cos(c + d*x))^(7/2), x, 5, -((2*a^2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*e^4*sqrt(cos(c + d*x)))) + (2*a^4*(e*cos(c + d*x))^(3/2))/(5*d*e^5*(a - a*sin(c + d*x))^2) + (2*a^4*(e*cos(c + d*x))^(3/2))/(5*d*e^5*(a^2 - a^2*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^2/(e*cos(c + d*x))^(9/2), x, 4, (2*a^2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(7*d*e^4*sqrt(e*cos(c + d*x))) + (2*a^2*sin(c + d*x))/(7*d*e^3*(e*cos(c + d*x))^(3/2)) + (4*(a^2 + a^2*sin(c + d*x)))/(7*d*e*(e*cos(c + d*x))^(7/2))]
    @test_int [(a + a*sin(c + d*x))^2/(e*cos(c + d*x))^(11/2), x, 5, -((2*a^2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(3*d*e^6*sqrt(cos(c + d*x)))) + (2*a^2*sin(c + d*x))/(9*d*e^3*(e*cos(c + d*x))^(5/2)) + (2*a^2*sin(c + d*x))/(3*d*e^5*sqrt(e*cos(c + d*x))) + (4*(a^2 + a^2*sin(c + d*x)))/(9*d*e*(e*cos(c + d*x))^(9/2))]

    @test_int [(e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))^3, x, 7, (-34*a^3*(e*cos(c + d*x))^(9/2))/(99*d*e) + (170*a^3*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(231*d*sqrt(e*cos(c + d*x))) + (170*a^3*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(231*d) + (34*a^3*e*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(77*d) - (2*a*(e*cos(c + d*x))^(9/2)*(a + a*sin(c + d*x))^2)/(13*d*e) - (34*(e*cos(c + d*x))^(9/2)*(a^3 + a^3*sin(c + d*x)))/(143*d*e)]
    @test_int [(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^3, x, 6, (-10*a^3*(e*cos(c + d*x))^(7/2))/(21*d*e) + (2*a^3*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(d*sqrt(cos(c + d*x))) + (2*a^3*e*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) - (2*a*(e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))^2)/(11*d*e) - (10*(e*cos(c + d*x))^(7/2)*(a^3 + a^3*sin(c + d*x)))/(33*d*e)]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^3, x, 6, (-26*a^3*(e*cos(c + d*x))^(5/2))/(35*d*e) + (26*a^3*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*d*sqrt(e*cos(c + d*x))) + (26*a^3*e*sqrt(e*cos(c + d*x))*sin(c + d*x))/(21*d) - (2*a*(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^2)/(9*d*e) - (26*(e*cos(c + d*x))^(5/2)*(a^3 + a^3*sin(c + d*x)))/(63*d*e)]
    @test_int [sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^3, x, 5, (-22*a^3*(e*cos(c + d*x))^(3/2))/(15*d*e) + (22*a^3*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*sqrt(cos(c + d*x))) - (2*a*(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^2)/(7*d*e) - (22*(e*cos(c + d*x))^(3/2)*(a^3 + a^3*sin(c + d*x)))/(35*d*e)]
    @test_int [(a + a*sin(c + d*x))^3/sqrt(e*cos(c + d*x)), x, 5, (-6*a^3*sqrt(e*cos(c + d*x)))/(d*e) + (6*a^3*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(d*sqrt(e*cos(c + d*x))) - (2*a*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^2)/(5*d*e) - (6*sqrt(e*cos(c + d*x))*(a^3 + a^3*sin(c + d*x)))/(5*d*e)]
    @test_int [(a + a*sin(c + d*x))^3/(e*cos(c + d*x))^(3/2), x, 5, (14*a^3*(e*cos(c + d*x))^(3/2))/(3*d*e^3) - (14*a^3*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*e^2*sqrt(cos(c + d*x))) + (4*a^5*(e*cos(c + d*x))^(7/2))/(d*e^5*(a - a*sin(c + d*x))^2)]
    @test_int [(a + a*sin(c + d*x))^3/(e*cos(c + d*x))^(5/2), x, 5, (10*a^3*sqrt(e*cos(c + d*x)))/(3*d*e^3) - (10*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*e^2*sqrt(e*cos(c + d*x))) + (4*a^5*(e*cos(c + d*x))^(5/2))/(3*d*e^5*(a - a*sin(c + d*x))^2)]
    @test_int [(a + a*sin(c + d*x))^3/(e*cos(c + d*x))^(7/2), x, 5, (6*a^3*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*e^4*sqrt(cos(c + d*x))) + (4*a^5*(e*cos(c + d*x))^(3/2))/(5*d*e^5*(a - a*sin(c + d*x))^2) - (6*a^6*(e*cos(c + d*x))^(3/2))/(5*d*e^5*(a^3 - a^3*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^3/(e*cos(c + d*x))^(9/2), x, 5, -((2*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*d*e^4*sqrt(e*cos(c + d*x)))) + (4*a^5*sqrt(e*cos(c + d*x)))/(7*d*e^5*(a - a*sin(c + d*x))^2) - (2*a^6*sqrt(e*cos(c + d*x)))/(21*d*e^5*(a^3 - a^3*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^3/(e*cos(c + d*x))^(11/2), x, 6, -((2*a^3*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(15*d*e^6*sqrt(cos(c + d*x)))) + (2*a^6*(e*cos(c + d*x))^(3/2))/(9*d*e^7*(a - a*sin(c + d*x))^3) + (2*a^5*(e*cos(c + d*x))^(3/2))/(15*d*e^7*(a - a*sin(c + d*x))^2) + (2*a^6*(e*cos(c + d*x))^(3/2))/(15*d*e^7*(a^3 - a^3*sin(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^4, x, 7, (-442*a^4*(e*cos(c + d*x))^(5/2))/(385*d*e) + (442*a^4*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(231*d*sqrt(e*cos(c + d*x))) + (442*a^4*e*sqrt(e*cos(c + d*x))*sin(c + d*x))/(231*d) - (2*a*(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^3)/(11*d*e) - (34*(e*cos(c + d*x))^(5/2)*(a^2 + a^2*sin(c + d*x))^2)/(99*d*e) - (442*(e*cos(c + d*x))^(5/2)*(a^4 + a^4*sin(c + d*x)))/(693*d*e)]
    @test_int [sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^4, x, 6, (-22*a^4*(e*cos(c + d*x))^(3/2))/(9*d*e) + (22*a^4*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(3*d*sqrt(cos(c + d*x))) - (2*a*(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^3)/(9*d*e) - (10*(e*cos(c + d*x))^(3/2)*(a^2 + a^2*sin(c + d*x))^2)/(21*d*e) - (22*(e*cos(c + d*x))^(3/2)*(a^4 + a^4*sin(c + d*x)))/(21*d*e)]
    @test_int [(a + a*sin(c + d*x))^4/sqrt(e*cos(c + d*x)), x, 6, (-78*a^4*sqrt(e*cos(c + d*x)))/(7*d*e) + (78*a^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(7*d*sqrt(e*cos(c + d*x))) - (2*a*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^3)/(7*d*e) - (26*sqrt(e*cos(c + d*x))*(a^2 + a^2*sin(c + d*x))^2)/(35*d*e) - (78*sqrt(e*cos(c + d*x))*(a^4 + a^4*sin(c + d*x)))/(35*d*e)]
    @test_int [(a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(3/2), x, 6, -((154*a^4*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*e^2*sqrt(cos(c + d*x)))) - (154*a^4*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(15*d*e^3) + (4*a^7*(e*cos(c + d*x))^(11/2))/(d*e^7*(a - a*sin(c + d*x))^3) + (44*a^8*(e*cos(c + d*x))^(7/2))/(3*d*e^5*(a^4 - a^4*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(5/2), x, 6, -((10*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*e^2*sqrt(e*cos(c + d*x)))) - (10*a^4*sqrt(e*cos(c + d*x))*sin(c + d*x))/(d*e^3) + (4*a^7*(e*cos(c + d*x))^(9/2))/(3*d*e^7*(a - a*sin(c + d*x))^3) + (12*a^8*(e*cos(c + d*x))^(5/2))/(d*e^5*(a^4 - a^4*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(7/2), x, 5, (42*a^4*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*e^4*sqrt(cos(c + d*x))) + (4*a^7*(e*cos(c + d*x))^(7/2))/(5*d*e^7*(a - a*sin(c + d*x))^3) - (28*a^8*(e*cos(c + d*x))^(3/2))/(5*d*e^5*(a^4 - a^4*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(9/2), x, 5, (10*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*d*e^4*sqrt(e*cos(c + d*x))) + (4*a^7*(e*cos(c + d*x))^(5/2))/(7*d*e^7*(a - a*sin(c + d*x))^3) - (20*a^8*sqrt(e*cos(c + d*x)))/(21*d*e^5*(a^4 - a^4*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(11/2), x, 6, (2*a^4*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(15*d*e^6*sqrt(cos(c + d*x))) + (4*a^7*(e*cos(c + d*x))^(3/2))/(9*d*e^7*(a - a*sin(c + d*x))^3) - (2*a^8*(e*cos(c + d*x))^(3/2))/(15*d*e^7*(a^2 - a^2*sin(c + d*x))^2) - (2*a^8*(e*cos(c + d*x))^(3/2))/(15*d*e^7*(a^4 - a^4*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^4/(e*cos(c + d*x))^(13/2), x, 6, -((2*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(77*d*e^6*sqrt(e*cos(c + d*x)))) + (4*a^7*sqrt(e*cos(c + d*x)))/(11*d*e^7*(a - a*sin(c + d*x))^3) - (2*a^8*sqrt(e*cos(c + d*x)))/(77*d*e^7*(a^2 - a^2*sin(c + d*x))^2) - (2*a^8*sqrt(e*cos(c + d*x)))/(77*d*e^7*(a^4 - a^4*sin(c + d*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(e*cos(c + d*x))^(11/2)/(a + a*sin(c + d*x)), x, 5, (2*e*(e*cos(c + d*x))^(9/2))/(9*a*d) + (10*e^6*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*a*d*sqrt(e*cos(c + d*x))) + (10*e^5*sqrt(e*cos(c + d*x))*sin(c + d*x))/(21*a*d) + (2*e^3*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*a*d)]
    @test_int [(e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x)), x, 4, (2*e*(e*cos(c + d*x))^(7/2))/(7*a*d) + (6*e^4*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*a*d*sqrt(cos(c + d*x))) + (2*e^3*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*a*d)]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x)), x, 4, (2*e*(e*cos(c + d*x))^(5/2))/(5*a*d) + (2*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*a*d*sqrt(e*cos(c + d*x))) + (2*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(3*a*d)]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + a*sin(c + d*x)), x, 3, (2*e*(e*cos(c + d*x))^(3/2))/(3*a*d) + (2*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(a*d*sqrt(cos(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + a*sin(c + d*x)), x, 3, (2*e*sqrt(e*cos(c + d*x)))/(a*d) + (2*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(a*d*sqrt(e*cos(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + a*sin(c + d*x)), x, 3, -((2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(a*d*sqrt(cos(c + d*x)))) - (2*(e*cos(c + d*x))^(3/2))/(d*e*(a + a*sin(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))), x, 3, (2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*a*d*sqrt(e*cos(c + d*x))) - (2*sqrt(e*cos(c + d*x)))/(3*d*e*(a + a*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))), x, 4, (-6*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*a*d*e^2*sqrt(cos(c + d*x))) + (6*sin(c + d*x))/(5*a*d*e*sqrt(e*cos(c + d*x))) - 2/(5*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))), x, 4, (10*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*a*d*e^2*sqrt(e*cos(c + d*x))) + (10*sin(c + d*x))/(21*a*d*e*(e*cos(c + d*x))^(3/2)) - 2/(7*d*e*(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))), x, 5, (-14*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(15*a*d*e^4*sqrt(cos(c + d*x))) + (14*sin(c + d*x))/(45*a*d*e*(e*cos(c + d*x))^(5/2)) + (14*sin(c + d*x))/(15*a*d*e^3*sqrt(e*cos(c + d*x))) - 2/(9*d*e*(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(11/2)/(a + a*sin(c + d*x))^2, x, 5, (6*e^6*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(7*a^2*d*sqrt(e*cos(c + d*x))) + (6*e^5*sqrt(e*cos(c + d*x))*sin(c + d*x))/(7*a^2*d) + (18*e^3*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(35*a^2*d) + (4*e*(e*cos(c + d*x))^(9/2))/(5*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x))^2, x, 4, (14*e^4*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*a^2*d*sqrt(cos(c + d*x))) + (14*e^3*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(15*a^2*d) + (4*e*(e*cos(c + d*x))^(7/2))/(3*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^2, x, 4, (10*e^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d*sqrt(e*cos(c + d*x))) + (10*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(3*a^2*d) + (4*e*(e*cos(c + d*x))^(5/2))/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + a*sin(c + d*x))^2, x, 3, (-6*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(a^2*d*sqrt(cos(c + d*x))) - (4*e*(e*cos(c + d*x))^(3/2))/(d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + a*sin(c + d*x))^2, x, 3, (-2*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*a^2*d*sqrt(e*cos(c + d*x))) - (4*e*sqrt(e*cos(c + d*x)))/(3*d*(a^2 + a^2*sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + a*sin(c + d*x))^2, x, 4, -((2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*a^2*d*sqrt(cos(c + d*x)))) - (2*(e*cos(c + d*x))^(3/2))/(5*d*e*(a + a*sin(c + d*x))^2) - (2*(e*cos(c + d*x))^(3/2))/(5*d*e*(a^2 + a^2*sin(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^2), x, 4, (2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(7*a^2*d*sqrt(e*cos(c + d*x))) - (2*sqrt(e*cos(c + d*x)))/(7*d*e*(a + a*sin(c + d*x))^2) - (2*sqrt(e*cos(c + d*x)))/(7*d*e*(a^2 + a^2*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^2), x, 5, (-2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(3*a^2*d*e^2*sqrt(cos(c + d*x))) + (2*sin(c + d*x))/(3*a^2*d*e*sqrt(e*cos(c + d*x))) - 2/(9*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^2) - 2/(9*d*e*sqrt(e*cos(c + d*x))*(a^2 + a^2*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^2), x, 5, (10*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(33*a^2*d*e^2*sqrt(e*cos(c + d*x))) + (10*sin(c + d*x))/(33*a^2*d*e*(e*cos(c + d*x))^(3/2)) - 2/(11*d*e*(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^2) - 2/(11*d*e*(e*cos(c + d*x))^(3/2)*(a^2 + a^2*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))^2), x, 6, (-42*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(65*a^2*d*e^4*sqrt(cos(c + d*x))) + (14*sin(c + d*x))/(65*a^2*d*e*(e*cos(c + d*x))^(5/2)) + (42*sin(c + d*x))/(65*a^2*d*e^3*sqrt(e*cos(c + d*x))) - 2/(13*d*e*(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^2) - 2/(13*d*e*(e*cos(c + d*x))^(5/2)*(a^2 + a^2*sin(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(15/2)/(a + a*sin(c + d*x))^3, x, 6, (26*e^3*(e*cos(c + d*x))^(9/2))/(45*a^3*d) + (26*e^8*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*a^3*d*sqrt(e*cos(c + d*x))) + (26*e^7*sqrt(e*cos(c + d*x))*sin(c + d*x))/(21*a^3*d) + (26*e^5*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(35*a^3*d) + (4*e*(e*cos(c + d*x))^(13/2))/(5*a*d*(a + a*sin(c + d*x))^2)]
    @test_int [(e*cos(c + d*x))^(13/2)/(a + a*sin(c + d*x))^3, x, 5, (22*e^3*(e*cos(c + d*x))^(7/2))/(21*a^3*d) + (22*e^6*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*a^3*d*sqrt(cos(c + d*x))) + (22*e^5*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(15*a^3*d) + (4*e*(e*cos(c + d*x))^(11/2))/(3*a*d*(a + a*sin(c + d*x))^2)]
    @test_int [(e*cos(c + d*x))^(11/2)/(a + a*sin(c + d*x))^3, x, 5, (18*e^3*(e*cos(c + d*x))^(5/2))/(5*a^3*d) + (6*e^6*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(a^3*d*sqrt(e*cos(c + d*x))) + (6*e^5*sqrt(e*cos(c + d*x))*sin(c + d*x))/(a^3*d) + (4*e*(e*cos(c + d*x))^(9/2))/(a*d*(a + a*sin(c + d*x))^2)]
    @test_int [(e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x))^3, x, 4, (-14*e^3*(e*cos(c + d*x))^(3/2))/(3*a^3*d) - (14*e^4*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(a^3*d*sqrt(cos(c + d*x))) - (4*e*(e*cos(c + d*x))^(7/2))/(a*d*(a + a*sin(c + d*x))^2)]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^3, x, 4, (-10*e^3*sqrt(e*cos(c + d*x)))/(3*a^3*d) - (10*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*a^3*d*sqrt(e*cos(c + d*x))) - (4*e*(e*cos(c + d*x))^(5/2))/(3*a*d*(a + a*sin(c + d*x))^2)]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + a*sin(c + d*x))^3, x, 4, (6*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*a^3*d*sqrt(cos(c + d*x))) - (4*e*(e*cos(c + d*x))^(3/2))/(5*a*d*(a + a*sin(c + d*x))^2) + (6*e*(e*cos(c + d*x))^(3/2))/(5*d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + a*sin(c + d*x))^3, x, 4, (-2*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*a^3*d*sqrt(e*cos(c + d*x))) - (4*e*sqrt(e*cos(c + d*x)))/(7*a*d*(a + a*sin(c + d*x))^2) + (2*e*sqrt(e*cos(c + d*x)))/(21*d*(a^3 + a^3*sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + a*sin(c + d*x))^3, x, 5, -((2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(15*a^3*d*sqrt(cos(c + d*x)))) - (2*(e*cos(c + d*x))^(3/2))/(9*d*e*(a + a*sin(c + d*x))^3) - (2*(e*cos(c + d*x))^(3/2))/(15*a*d*e*(a + a*sin(c + d*x))^2) - (2*(e*cos(c + d*x))^(3/2))/(15*d*e*(a^3 + a^3*sin(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^3), x, 5, (10*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(77*a^3*d*sqrt(e*cos(c + d*x))) - (2*sqrt(e*cos(c + d*x)))/(11*d*e*(a + a*sin(c + d*x))^3) - (10*sqrt(e*cos(c + d*x)))/(77*a*d*e*(a + a*sin(c + d*x))^2) - (10*sqrt(e*cos(c + d*x)))/(77*d*e*(a^3 + a^3*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^3), x, 6, (-14*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(39*a^3*d*e^2*sqrt(cos(c + d*x))) + (14*sin(c + d*x))/(39*a^3*d*e*sqrt(e*cos(c + d*x))) - 2/(13*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^3) - 14/(117*a*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^2) - 14/(117*d*e*sqrt(e*cos(c + d*x))*(a^3 + a^3*sin(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(15/2)/(a + a*sin(c + d*x))^4, x, 6, (78*e^8*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(7*a^4*d*sqrt(e*cos(c + d*x))) + (78*e^7*sqrt(e*cos(c + d*x))*sin(c + d*x))/(7*a^4*d) + (234*e^5*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(35*a^4*d) + (4*e*(e*cos(c + d*x))^(13/2))/(a*d*(a + a*sin(c + d*x))^3) + (52*e^3*(e*cos(c + d*x))^(9/2))/(5*d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(13/2)/(a + a*sin(c + d*x))^4, x, 5, -((154*e^6*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*a^4*d*sqrt(cos(c + d*x)))) - (154*e^5*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(15*a^4*d) - (4*e*(e*cos(c + d*x))^(11/2))/(a*d*(a + a*sin(c + d*x))^3) - (44*e^3*(e*cos(c + d*x))^(7/2))/(3*d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(11/2)/(a + a*sin(c + d*x))^4, x, 5, -((10*e^6*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(a^4*d*sqrt(e*cos(c + d*x)))) - (10*e^5*sqrt(e*cos(c + d*x))*sin(c + d*x))/(a^4*d) - (4*e*(e*cos(c + d*x))^(9/2))/(3*a*d*(a + a*sin(c + d*x))^3) - (12*e^3*(e*cos(c + d*x))^(5/2))/(d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x))^4, x, 4, (42*e^4*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*a^4*d*sqrt(cos(c + d*x))) - (4*e*(e*cos(c + d*x))^(7/2))/(5*a*d*(a + a*sin(c + d*x))^3) + (28*e^3*(e*cos(c + d*x))^(3/2))/(5*d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^4, x, 4, (10*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*a^4*d*sqrt(e*cos(c + d*x))) - (4*e*(e*cos(c + d*x))^(5/2))/(7*a*d*(a + a*sin(c + d*x))^3) + (20*e^3*sqrt(e*cos(c + d*x)))/(21*d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + a*sin(c + d*x))^4, x, 5, (2*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(15*a^4*d*sqrt(cos(c + d*x))) - (4*e*(e*cos(c + d*x))^(3/2))/(9*a*d*(a + a*sin(c + d*x))^3) + (2*e*(e*cos(c + d*x))^(3/2))/(15*d*(a^2 + a^2*sin(c + d*x))^2) + (2*e*(e*cos(c + d*x))^(3/2))/(15*d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + a*sin(c + d*x))^4, x, 5, (-2*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(77*a^4*d*sqrt(e*cos(c + d*x))) - (4*e*sqrt(e*cos(c + d*x)))/(11*a*d*(a + a*sin(c + d*x))^3) + (2*e*sqrt(e*cos(c + d*x)))/(77*d*(a^2 + a^2*sin(c + d*x))^2) + (2*e*sqrt(e*cos(c + d*x)))/(77*d*(a^4 + a^4*sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + a*sin(c + d*x))^4, x, 6, -((2*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(39*a^4*d*sqrt(cos(c + d*x)))) - (2*(e*cos(c + d*x))^(3/2))/(13*d*e*(a + a*sin(c + d*x))^4) - (10*(e*cos(c + d*x))^(3/2))/(117*a*d*e*(a + a*sin(c + d*x))^3) - (2*(e*cos(c + d*x))^(3/2))/(39*d*e*(a^2 + a^2*sin(c + d*x))^2) - (2*(e*cos(c + d*x))^(3/2))/(39*d*e*(a^4 + a^4*sin(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^4), x, 6, (2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(33*a^4*d*sqrt(e*cos(c + d*x))) - (2*sqrt(e*cos(c + d*x)))/(15*d*e*(a + a*sin(c + d*x))^4) - (14*sqrt(e*cos(c + d*x)))/(165*a*d*e*(a + a*sin(c + d*x))^3) - (2*sqrt(e*cos(c + d*x)))/(33*d*e*(a^2 + a^2*sin(c + d*x))^2) - (2*sqrt(e*cos(c + d*x)))/(33*d*e*(a^4 + a^4*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^4), x, 7, (-42*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(221*a^4*d*e^2*sqrt(cos(c + d*x))) + (42*sin(c + d*x))/(221*a^4*d*e*sqrt(e*cos(c + d*x))) - 2/(17*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^4) - 18/(221*a*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^3) - 14/(221*d*e*sqrt(e*cos(c + d*x))*(a^2 + a^2*sin(c + d*x))^2) - 14/(221*d*e*sqrt(e*cos(c + d*x))*(a^4 + a^4*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^(p/2)*(a+a*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(e*cos(c + d*x))^(3/2)*sqrt(a + a*sin(c + d*x)), x, 8, -((a*(e*cos(c + d*x))^(5/2))/(2*d*e*sqrt(a + a*sin(c + d*x)))) + (3*e*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d) - (3*e^(3/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*(1 + cos(c + d*x) + sin(c + d*x))) + (3*e^(3/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)), x, 7, -((a*(e*cos(c + d*x))^(3/2))/(d*e*sqrt(a + a*sin(c + d*x)))) + (sqrt(e)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*(1 + cos(c + d*x) + sin(c + d*x))) + (sqrt(e)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [sqrt(a + a*sin(c + d*x))/sqrt(e*cos(c + d*x)), x, 6, -((2*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*sqrt(e)*(1 + cos(c + d*x) + sin(c + d*x)))) + (2*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*sqrt(e)*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [sqrt(a + a*sin(c + d*x))/(e*cos(c + d*x))^(3/2), x, 1, (2*sqrt(a + a*sin(c + d*x)))/(d*e*sqrt(e*cos(c + d*x)))]
    @test_int [sqrt(a + a*sin(c + d*x))/(e*cos(c + d*x))^(5/2), x, 2, -((2*sqrt(a + a*sin(c + d*x)))/(d*e*(e*cos(c + d*x))^(3/2))) + (4*(a + a*sin(c + d*x))^(3/2))/(3*a*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [sqrt(a + a*sin(c + d*x))/(e*cos(c + d*x))^(7/2), x, 3, -((2*sqrt(a + a*sin(c + d*x)))/(3*d*e*(e*cos(c + d*x))^(5/2))) + (8*(a + a*sin(c + d*x))^(3/2))/(3*a*d*e*(e*cos(c + d*x))^(5/2)) - (16*(a + a*sin(c + d*x))^(5/2))/(15*a^2*d*e*(e*cos(c + d*x))^(5/2))]
    @test_int [sqrt(a + a*sin(c + d*x))/(e*cos(c + d*x))^(9/2), x, 4, -((2*sqrt(a + a*sin(c + d*x)))/(5*d*e*(e*cos(c + d*x))^(7/2))) - (12*(a + a*sin(c + d*x))^(3/2))/(5*a*d*e*(e*cos(c + d*x))^(7/2)) + (16*(a + a*sin(c + d*x))^(5/2))/(5*a^2*d*e*(e*cos(c + d*x))^(7/2)) - (32*(a + a*sin(c + d*x))^(7/2))/(35*a^3*d*e*(e*cos(c + d*x))^(7/2))]

    @test_int [(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^(3/2), x, 10, -((15*a^3*(e*cos(c + d*x))^(7/2))/(32*d*e*(a + a*sin(c + d*x))^(3/2))) + (15*a^2*e*(e*cos(c + d*x))^(3/2))/(64*d*sqrt(a + a*sin(c + d*x))) - (3*a^2*(e*cos(c + d*x))^(7/2))/(8*d*e*sqrt(a + a*sin(c + d*x))) - (a*(e*cos(c + d*x))^(7/2)*sqrt(a + a*sin(c + d*x)))/(4*d*e) + (45*a*e^(5/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(64*d*(1 + cos(c + d*x) + sin(c + d*x))) + (45*a*e^(5/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(64*d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(3/2), x, 9, -((7*a^2*(e*cos(c + d*x))^(5/2))/(12*d*e*sqrt(a + a*sin(c + d*x)))) + (7*a*e*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(8*d) - (a*(e*cos(c + d*x))^(5/2)*sqrt(a + a*sin(c + d*x)))/(3*d*e) - (7*a*e^(3/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(8*d*(1 + cos(c + d*x) + sin(c + d*x))) + (7*a*e^(3/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(8*d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^(3/2), x, 8, -((5*a^2*(e*cos(c + d*x))^(3/2))/(4*d*e*sqrt(a + a*sin(c + d*x)))) - (a*(e*cos(c + d*x))^(3/2)*sqrt(a + a*sin(c + d*x)))/(2*d*e) + (5*a*sqrt(e)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*(1 + cos(c + d*x) + sin(c + d*x))) + (5*a*sqrt(e)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^(3/2)/sqrt(e*cos(c + d*x)), x, 7, -((a*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*e)) - (3*a*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*sqrt(e)*(1 + cos(c + d*x) + sin(c + d*x))) + (3*a*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*sqrt(e)*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(3/2), x, 7, (4*a*sqrt(a + a*sin(c + d*x)))/(d*e*sqrt(e*cos(c + d*x))) - (2*a^2*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*e^(3/2)*(a + a*cos(c + d*x) + a*sin(c + d*x))) - (2*a^2*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*e^(3/2)*(a + a*cos(c + d*x) + a*sin(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(5/2), x, 1, (2*(a + a*sin(c + d*x))^(3/2))/(3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + a*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(7/2), x, 2, (2*(a + a*sin(c + d*x))^(3/2))/(d*e*(e*cos(c + d*x))^(5/2)) - (4*(a + a*sin(c + d*x))^(5/2))/(5*a*d*e*(e*cos(c + d*x))^(5/2))]
    @test_int [(a + a*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(9/2), x, 3, -((2*(a + a*sin(c + d*x))^(3/2))/(d*e*(e*cos(c + d*x))^(7/2))) + (8*(a + a*sin(c + d*x))^(5/2))/(3*a*d*e*(e*cos(c + d*x))^(7/2)) - (16*(a + a*sin(c + d*x))^(7/2))/(21*a^2*d*e*(e*cos(c + d*x))^(7/2))]
    @test_int [(a + a*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(11/2), x, 4, -((2*(a + a*sin(c + d*x))^(3/2))/(3*d*e*(e*cos(c + d*x))^(9/2))) + (4*(a + a*sin(c + d*x))^(5/2))/(a*d*e*(e*cos(c + d*x))^(9/2)) - (16*(a + a*sin(c + d*x))^(7/2))/(5*a^2*d*e*(e*cos(c + d*x))^(9/2)) + (32*(a + a*sin(c + d*x))^(9/2))/(45*a^3*d*e*(e*cos(c + d*x))^(9/2))]

    @test_int [(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(5/2), x, 10, -((77*a^3*(e*cos(c + d*x))^(5/2))/(96*d*e*sqrt(a + a*sin(c + d*x)))) + (77*a^2*e*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(64*d) - (11*a^2*(e*cos(c + d*x))^(5/2)*sqrt(a + a*sin(c + d*x)))/(24*d*e) - (77*a^2*e^(3/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(64*d*(1 + cos(c + d*x) + sin(c + d*x))) + (77*a^2*e^(3/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(64*d*(1 + cos(c + d*x) + sin(c + d*x))) - (a*(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^(3/2))/(4*d*e)]
    @test_int [sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^(5/2), x, 9, -((15*a^3*(e*cos(c + d*x))^(3/2))/(8*d*e*sqrt(a + a*sin(c + d*x)))) - (3*a^2*(e*cos(c + d*x))^(3/2)*sqrt(a + a*sin(c + d*x)))/(4*d*e) + (15*a^2*sqrt(e)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(8*d*(1 + cos(c + d*x) + sin(c + d*x))) + (15*a^2*sqrt(e)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(8*d*(1 + cos(c + d*x) + sin(c + d*x))) - (a*(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(3/2))/(3*d*e)]
    @test_int [(a + a*sin(c + d*x))^(5/2)/sqrt(e*cos(c + d*x)), x, 8, -((7*a^2*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*e)) - (21*a^2*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*sqrt(e)*(1 + cos(c + d*x) + sin(c + d*x))) + (21*a^2*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*sqrt(e)*(1 + cos(c + d*x) + sin(c + d*x))) - (a*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^(3/2))/(2*d*e)]
    @test_int [(a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(3/2), x, 8, (5*a^3*(e*cos(c + d*x))^(3/2))/(d*e^3*sqrt(a + a*sin(c + d*x))) - (5*a^2*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*e^(3/2)*(1 + cos(c + d*x) + sin(c + d*x))) - (5*a^2*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*e^(3/2)*(1 + cos(c + d*x) + sin(c + d*x))) + (4*a*(a + a*sin(c + d*x))^(3/2))/(d*e*sqrt(e*cos(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(5/2), x, 7, (2*a^2*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*e^(5/2)*(1 + cos(c + d*x) + sin(c + d*x))) - (2*a^2*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*e^(5/2)*(1 + cos(c + d*x) + sin(c + d*x))) + (4*a*(a + a*sin(c + d*x))^(3/2))/(3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(7/2), x, 1, (2*(a + a*sin(c + d*x))^(5/2))/(5*d*e*(e*cos(c + d*x))^(5/2))]
    @test_int [(a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(9/2), x, 2, (2*(a + a*sin(c + d*x))^(5/2))/(3*d*e*(e*cos(c + d*x))^(7/2)) - (4*(a + a*sin(c + d*x))^(7/2))/(21*a*d*e*(e*cos(c + d*x))^(7/2))]
    @test_int [(a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(11/2), x, 3, (2*(a + a*sin(c + d*x))^(5/2))/(d*e*(e*cos(c + d*x))^(9/2)) - (8*(a + a*sin(c + d*x))^(7/2))/(5*a*d*e*(e*cos(c + d*x))^(9/2)) + (16*(a + a*sin(c + d*x))^(9/2))/(45*a^2*d*e*(e*cos(c + d*x))^(9/2))]
    @test_int [(a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(13/2), x, 4, -((2*(a + a*sin(c + d*x))^(5/2))/(d*e*(e*cos(c + d*x))^(11/2))) + (4*(a + a*sin(c + d*x))^(7/2))/(a*d*e*(e*cos(c + d*x))^(11/2)) - (16*(a + a*sin(c + d*x))^(9/2))/(7*a^2*d*e*(e*cos(c + d*x))^(11/2)) + (32*(a + a*sin(c + d*x))^(11/2))/(77*a^3*d*e*(e*cos(c + d*x))^(11/2))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(e*cos(c + d*x))^(5/2)/sqrt(a + a*sin(c + d*x)), x, 8, -((a*(e*cos(c + d*x))^(7/2))/(2*d*e*(a + a*sin(c + d*x))^(3/2))) + (e*(e*cos(c + d*x))^(3/2))/(4*d*sqrt(a + a*sin(c + d*x))) + (3*e^(5/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*(a + a*cos(c + d*x) + a*sin(c + d*x))) + (3*e^(5/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*(a + a*cos(c + d*x) + a*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/sqrt(a + a*sin(c + d*x)), x, 7, (e*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a*d) - (e^(3/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a*d*(1 + cos(c + d*x) + sin(c + d*x))) + (e^(3/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a*d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/sqrt(a + a*sin(c + d*x)), x, 6, (2*sqrt(e)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*(a + a*cos(c + d*x) + a*sin(c + d*x))) + (2*sqrt(e)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*(a + a*cos(c + d*x) + a*sin(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x))), x, 1, (-2*sqrt(e*cos(c + d*x)))/(d*e*sqrt(a + a*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*sqrt(a + a*sin(c + d*x))), x, 2, -(2/(3*d*e*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))) + (4*sqrt(a + a*sin(c + d*x)))/(3*a*d*e*sqrt(e*cos(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(5/2)*sqrt(a + a*sin(c + d*x))), x, 3, -(2/(5*d*e*(e*cos(c + d*x))^(3/2)*sqrt(a + a*sin(c + d*x)))) - (8*sqrt(a + a*sin(c + d*x)))/(5*a*d*e*(e*cos(c + d*x))^(3/2)) + (16*(a + a*sin(c + d*x))^(3/2))/(15*a^2*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [1/((e*cos(c + d*x))^(7/2)*sqrt(a + a*sin(c + d*x))), x, 4, -(2/(7*d*e*(e*cos(c + d*x))^(5/2)*sqrt(a + a*sin(c + d*x)))) - (4*sqrt(a + a*sin(c + d*x)))/(7*a*d*e*(e*cos(c + d*x))^(5/2)) + (16*(a + a*sin(c + d*x))^(3/2))/(7*a^2*d*e*(e*cos(c + d*x))^(5/2)) - (32*(a + a*sin(c + d*x))^(5/2))/(35*a^3*d*e*(e*cos(c + d*x))^(5/2))]

    @test_int [(e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^(3/2), x, 8, (e*(e*cos(c + d*x))^(5/2))/(2*a*d*sqrt(a + a*sin(c + d*x))) + (5*e^3*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*a^2*d) - (5*e^(7/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*a^2*d*(1 + cos(c + d*x) + sin(c + d*x))) + (5*e^(7/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*a^2*d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + a*sin(c + d*x))^(3/2), x, 7, (e*(e*cos(c + d*x))^(3/2))/(a*d*sqrt(a + a*sin(c + d*x))) + (3*e^(5/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*(a^2 + a^2*cos(c + d*x) + a^2*sin(c + d*x))) + (3*e^(5/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*(a^2 + a^2*cos(c + d*x) + a^2*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + a*sin(c + d*x))^(3/2), x, 8, -((2*(e*cos(c + d*x))^(5/2))/(d*e*(a + a*sin(c + d*x))^(3/2))) - (2*e*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a^2*d) + (2*e^(3/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a^2*d*(1 + cos(c + d*x) + sin(c + d*x))) - (2*e^(3/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a^2*d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + a*sin(c + d*x))^(3/2), x, 1, (-2*(e*cos(c + d*x))^(3/2))/(3*d*e*(a + a*sin(c + d*x))^(3/2))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^(3/2)), x, 2, (-2*sqrt(e*cos(c + d*x)))/(5*d*e*(a + a*sin(c + d*x))^(3/2)) - (4*sqrt(e*cos(c + d*x)))/(5*a*d*e*sqrt(a + a*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(3/2)), x, 3, -(2/(7*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^(3/2))) - 8/(21*a*d*e*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x))) + (16*sqrt(a + a*sin(c + d*x)))/(21*a^2*d*e*sqrt(e*cos(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^(3/2)), x, 4, -(2/(9*d*e*(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(3/2))) - 4/(15*a*d*e*(e*cos(c + d*x))^(3/2)*sqrt(a + a*sin(c + d*x))) - (16*sqrt(a + a*sin(c + d*x)))/(15*a^2*d*e*(e*cos(c + d*x))^(3/2)) + (32*(a + a*sin(c + d*x))^(3/2))/(45*a^3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [1/((e*cos(c + d*x))^(7/2)*(a + a*sin(c + d*x))^(3/2)), x, 5, -(2/(11*d*e*(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^(3/2))) - 16/(77*a*d*e*(e*cos(c + d*x))^(5/2)*sqrt(a + a*sin(c + d*x))) - (32*sqrt(a + a*sin(c + d*x)))/(77*a^2*d*e*(e*cos(c + d*x))^(5/2)) + (128*(a + a*sin(c + d*x))^(3/2))/(77*a^3*d*e*(e*cos(c + d*x))^(5/2)) - (256*(a + a*sin(c + d*x))^(5/2))/(385*a^4*d*e*(e*cos(c + d*x))^(5/2))]

    @test_int [(e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x))^(5/2), x, 9, (e*(e*cos(c + d*x))^(7/2))/(2*a*d*(a + a*sin(c + d*x))^(3/2)) + (7*e^3*(e*cos(c + d*x))^(3/2))/(4*a^2*d*sqrt(a + a*sin(c + d*x))) + (21*e^(9/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*(a^3 + a^3*cos(c + d*x) + a^3*sin(c + d*x))) + (21*e^(9/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(4*d*(a^3 + a^3*cos(c + d*x) + a^3*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^(5/2), x, 8, -((4*e*(e*cos(c + d*x))^(5/2))/(a*d*(a + a*sin(c + d*x))^(3/2))) - (5*e^3*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a^3*d) + (5*e^(7/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a^3*d*(1 + cos(c + d*x) + sin(c + d*x))) - (5*e^(7/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(a^3*d*(1 + cos(c + d*x) + sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + a*sin(c + d*x))^(5/2), x, 7, -((4*e*(e*cos(c + d*x))^(3/2))/(3*a*d*(a + a*sin(c + d*x))^(3/2))) - (2*e^(5/2)*asinh(sqrt(e*cos(c + d*x))/sqrt(e))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*(a^3 + a^3*cos(c + d*x) + a^3*sin(c + d*x))) - (2*e^(5/2)*atan((sqrt(e)*sin(c + d*x))/(sqrt(e*cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*sqrt(a + a*sin(c + d*x)))/(d*(a^3 + a^3*cos(c + d*x) + a^3*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + a*sin(c + d*x))^(5/2), x, 1, (-2*(e*cos(c + d*x))^(5/2))/(5*d*e*(a + a*sin(c + d*x))^(5/2))]
    @test_int [sqrt(e*cos(c + d*x))/(a + a*sin(c + d*x))^(5/2), x, 2, (-2*(e*cos(c + d*x))^(3/2))/(7*d*e*(a + a*sin(c + d*x))^(5/2)) - (4*(e*cos(c + d*x))^(3/2))/(21*a*d*e*(a + a*sin(c + d*x))^(3/2))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^(5/2)), x, 3, (-2*sqrt(e*cos(c + d*x)))/(9*d*e*(a + a*sin(c + d*x))^(5/2)) - (8*sqrt(e*cos(c + d*x)))/(45*a*d*e*(a + a*sin(c + d*x))^(3/2)) - (16*sqrt(e*cos(c + d*x)))/(45*a^2*d*e*sqrt(a + a*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(5/2)), x, 4, -(2/(11*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^(5/2))) - 12/(77*a*d*e*sqrt(e*cos(c + d*x))*(a + a*sin(c + d*x))^(3/2)) - 16/(77*a^2*d*e*sqrt(e*cos(c + d*x))*sqrt(a + a*sin(c + d*x))) + (32*sqrt(a + a*sin(c + d*x)))/(77*a^3*d*e*sqrt(e*cos(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^(5/2)), x, 5, -(2/(13*d*e*(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(5/2))) - 16/(117*a*d*e*(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(3/2)) - 32/(195*a^2*d*e*(e*cos(c + d*x))^(3/2)*sqrt(a + a*sin(c + d*x))) - (128*sqrt(a + a*sin(c + d*x)))/(195*a^3*d*e*(e*cos(c + d*x))^(3/2)) + (256*(a + a*sin(c + d*x))^(3/2))/(585*a^4*d*e*(e*cos(c + d*x))^(3/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^(p/3)*(a+a*sin(e+f*x))^(m/2)=#

    @test_int [(e*cos(c + d*x))^(7/3)/sqrt(a + a*sin(c + d*x)), x, 3, -((3*2^(1/6)*a*(e*cos(c + d*x))^(10/3)*HypergeometricFunctions._₂F₁(-(1/6), 5/3, 8/3, (1/2)*(1 - sin(c + d*x))))/(5*d*e*(1 + sin(c + d*x))^(1/6)*(a + a*sin(c + d*x))^(3/2)))]
    @test_int [(e*cos(c + d*x))^(5/3)/sqrt(a + a*sin(c + d*x)), x, 3, -((3*a*(e*cos(c + d*x))^(8/3)*HypergeometricFunctions._₂F₁(1/6, 4/3, 7/3, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(1/6))/(4*2^(1/6)*d*e*(a + a*sin(c + d*x))^(3/2)))]
    @test_int [(e*cos(c + d*x))^(2/3)/sqrt(a + a*sin(c + d*x)), x, 3, -((3*2^(1/3)*a*(e*cos(c + d*x))^(5/3)*HypergeometricFunctions._₂F₁(2/3, 5/6, 11/6, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(2/3))/(5*d*e*(a + a*sin(c + d*x))^(3/2)))]
    @test_int [(e*cos(c + d*x))^(1/3)/sqrt(a + a*sin(c + d*x)), x, 3, -((3*a*(e*cos(c + d*x))^(4/3)*HypergeometricFunctions._₂F₁(2/3, 5/6, 5/3, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(5/6))/(2*2^(5/6)*d*e*(a + a*sin(c + d*x))^(3/2)))]
    @test_int [1/(e*cos(c + d*x))^(1/3)/sqrt(a + a*sin(c + d*x)), x, 3, -((3*(e*cos(c + d*x))^(2/3)*HypergeometricFunctions._₂F₁(1/3, 7/6, 4/3, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(1/6))/(2*2^(1/6)*d*e*sqrt(a + a*sin(c + d*x))))]
    @test_int [1/(e*cos(c + d*x))^(4/3)/sqrt(a + a*sin(c + d*x)), x, 3, (3*HypergeometricFunctions._₂F₁(-(1/6), 5/3, 5/6, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(2/3))/(2^(2/3)*d*e*(e*cos(c + d*x))^(1/3)*sqrt(a + a*sin(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*when*p*symbolic=#

    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^8, x, 2, -((2^(17/2 + p/2)*a^8*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-15 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - p)))/(d*e*(1 + p)))]
    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^3, x, 2, -((2^(7/2 + p/2)*a^3*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-5 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - p)))/(d*e*(1 + p)))]
    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^2, x, 2, -((2^(5/2 + p/2)*a^2*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-3 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - p)))/(d*e*(1 + p)))]
    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^1, x, 2, -((2^(3/2 + p/2)*a*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-1 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - p)))/(d*e*(1 + p)))]
    @test_int [(e*cos(c + d*x))^p/(a + a*sin(c + d*x))^1, x, 2, -((2^(-(1/2) + p/2)*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((3 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - p)))/(a*d*e*(1 + p)))]
    @test_int [(e*cos(c + d*x))^p/(a + a*sin(c + d*x))^2, x, 2, -((2^((1/2)*(-3 + p))*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((5 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - p)))/(a^2*d*e*(1 + p)))]
    @test_int [(e*cos(c + d*x))^p/(a + a*sin(c + d*x))^3, x, 2, -((2^((1/2)*(-5 + p))*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((7 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - p)))/(a^3*d*e*(1 + p)))]
    @test_int [(e*cos(c + d*x))^p/(a + a*sin(c + d*x))^8, x, 2, -((2^((1/2)*(-15 + p))*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((17 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - p)))/(a^8*d*e*(1 + p)))]

    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^(7/2), x, 3, -((2^(4 + p/2)*a^4*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-6 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x))))/((1 + sin(c + d*x))^(p/2)*(d*e*(1 + p)*sqrt(a + a*sin(c + d*x)))))]
    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^(5/2), x, 3, -((2^(3 + p/2)*a^3*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-4 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x))))/((1 + sin(c + d*x))^(p/2)*(d*e*(1 + p)*sqrt(a + a*sin(c + d*x)))))]
    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^(3/2), x, 3, -((2^(2 + p/2)*a^2*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(-2 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x))))/((1 + sin(c + d*x))^(p/2)*(d*e*(1 + p)*sqrt(a + a*sin(c + d*x)))))]
    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^(1/2), x, 3, -((2^(1 + p/2)*a*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁(-(p/2), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x))))/((1 + sin(c + d*x))^(p/2)*(d*e*(1 + p)*sqrt(a + a*sin(c + d*x)))))]
    @test_int [(e*cos(c + d*x))^p/(a + a*sin(c + d*x))^(1/2), x, 3, -((2^(p/2)*a*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((2 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(1 - p/2))/(d*e*(1 + p)*(a + a*sin(c + d*x))^(3/2)))]
    @test_int [(e*cos(c + d*x))^p/(a + a*sin(c + d*x))^(3/2), x, 3, -((2^(-1 + p/2)*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((4 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(1 - p/2))/(d*e*(1 + p)*(a + a*sin(c + d*x))^(3/2)))]
    @test_int [(e*cos(c + d*x))^p/(a + a*sin(c + d*x))^(5/2), x, 3, -((2^(-2 + p/2)*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((6 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(1 - p/2))/(a*d*e*(1 + p)*(a + a*sin(c + d*x))^(3/2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*when*m*symbolic=#

    @test_int [(e*cos(c + d*x))^p*(a + a*sin(c + d*x))^m, x, 3, -((2^(1/2 + m + p/2)*a*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁((1/2)*(1 - 2*m - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(1 - 2*m - p))*(a + a*sin(c + d*x))^(-1 + m))/(d*e*(1 + p)))]

    @test_int [cos(c + d*x)^7*(a + a*sin(c + d*x))^m, x, 3, (8*(a + a*sin(c + d*x))^(4 + m))/(a^4*d*(4 + m)) - (12*(a + a*sin(c + d*x))^(5 + m))/(a^5*d*(5 + m)) + (6*(a + a*sin(c + d*x))^(6 + m))/(a^6*d*(6 + m)) - (a + a*sin(c + d*x))^(7 + m)/(a^7*d*(7 + m))]
    @test_int [cos(c + d*x)^5*(a + a*sin(c + d*x))^m, x, 3, (4*(a + a*sin(c + d*x))^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*sin(c + d*x))^(4 + m))/(a^4*d*(4 + m)) + (a + a*sin(c + d*x))^(5 + m)/(a^5*d*(5 + m))]
    @test_int [cos(c + d*x)^3*(a + a*sin(c + d*x))^m, x, 3, (2*(a + a*sin(c + d*x))^(2 + m))/(a^2*d*(2 + m)) - (a + a*sin(c + d*x))^(3 + m)/(a^3*d*(3 + m))]
    @test_int [cos(c + d*x)^1*(a + a*sin(c + d*x))^m, x, 2, (a + a*sin(c + d*x))^(1 + m)/(a*d*(1 + m))]
    @test_int [sec(c + d*x)^1*(a + a*sin(c + d*x))^m, x, 2, (HypergeometricFunctions._₂F₁(1, m, 1 + m, (1/2)*(1 + sin(c + d*x)))*(a + a*sin(c + d*x))^m)/(2*d*m)]
    @test_int [sec(c + d*x)^3*(a + a*sin(c + d*x))^m, x, 2, -((a*HypergeometricFunctions._₂F₁(2, -1 + m, m, (1/2)*(1 + sin(c + d*x)))*(a + a*sin(c + d*x))^(-1 + m))/(4*d*(1 - m)))]
    @test_int [sec(c + d*x)^5*(a + a*sin(c + d*x))^m, x, 2, -((a^2*HypergeometricFunctions._₂F₁(3, -2 + m, -1 + m, (1/2)*(1 + sin(c + d*x)))*(a + a*sin(c + d*x))^(-2 + m))/(8*d*(2 - m)))]

    @test_int [cos(c + d*x)^4*(a + a*sin(c + d*x))^m, x, 3, -((2^(5/2 + m)*a^2*cos(c + d*x)^5*HypergeometricFunctions._₂F₁(5/2, -(3/2) - m, 7/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - m)*(a + a*sin(c + d*x))^(-2 + m))/(5*d))]
    @test_int [cos(c + d*x)^2*(a + a*sin(c + d*x))^m, x, 3, -((2^(3/2 + m)*a*cos(c + d*x)^3*HypergeometricFunctions._₂F₁(3/2, -(1/2) - m, 5/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/2) - m)*(a + a*sin(c + d*x))^(-1 + m))/(3*d))]
    @test_int [sec(c + d*x)^2*(a + a*sin(c + d*x))^m, x, 3, (2^(-(1/2) + m)*HypergeometricFunctions._₂F₁(-(1/2), 3/2 - m, 1/2, (1/2)*(1 - sin(c + d*x)))*sec(c + d*x)*(1 + sin(c + d*x))^(1/2 - m)*(a + a*sin(c + d*x))^m)/d]
    @test_int [sec(c + d*x)^4*(a + a*sin(c + d*x))^m, x, 3, (2^(-(3/2) + m)*HypergeometricFunctions._₂F₁(-(3/2), 5/2 - m, -(1/2), (1/2)*(1 - sin(c + d*x)))*sec(c + d*x)^3*(1 + sin(c + d*x))^(1/2 - m)*(a + a*sin(c + d*x))^(1 + m))/(3*a*d)]

    @test_int [(e*cos(c + d*x))^(5/2)*(a + a*sin(c + d*x))^m, x, 3, -((2^(11/4 + m)*a*(e*cos(c + d*x))^(7/2)*HypergeometricFunctions._₂F₁(7/4, -(3/4) - m, 11/4, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(3/4) - m)*(a + a*sin(c + d*x))^(-1 + m))/(7*d*e))]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^m, x, 3, -((2^(9/4 + m)*a*(e*cos(c + d*x))^(5/2)*HypergeometricFunctions._₂F₁(5/4, -(1/4) - m, 9/4, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1/4) - m)*(a + a*sin(c + d*x))^(-1 + m))/(5*d*e))]
    @test_int [(e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^m, x, 3, -((2^(7/4 + m)*a*(e*cos(c + d*x))^(3/2)*HypergeometricFunctions._₂F₁(3/4, 1/4 - m, 7/4, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(1/4 - m)*(a + a*sin(c + d*x))^(-1 + m))/(3*d*e))]
    @test_int [(a + a*sin(c + d*x))^m/(e*cos(c + d*x))^(1/2), x, 3, -((2^(5/4 + m)*a*sqrt(e*cos(c + d*x))*HypergeometricFunctions._₂F₁(1/4, 3/4 - m, 5/4, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(3/4 - m)*(a + a*sin(c + d*x))^(-1 + m))/(d*e))]
    @test_int [(a + a*sin(c + d*x))^m/(e*cos(c + d*x))^(3/2), x, 3, (2^(3/4 + m)*HypergeometricFunctions._₂F₁(-(1/4), 5/4 - m, 3/4, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(1/4 - m)*(a + a*sin(c + d*x))^m)/(d*e*sqrt(e*cos(c + d*x)))]
    @test_int [(a + a*sin(c + d*x))^m/(e*cos(c + d*x))^(5/2), x, 3, (2^(1/4 + m)*HypergeometricFunctions._₂F₁(-(3/4), 7/4 - m, 1/4, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(3/4 - m)*(a + a*sin(c + d*x))^m)/(3*d*e*(e*cos(c + d*x))^(3/2))]

    @test_int [(e*cos(c + d*x))^(-4 - m)*(a + a*sin(c + d*x))^m, x, 4, -(((e*cos(c + d*x))^(-3 - m)*(a + a*sin(c + d*x))^m)/(d*e*(3 - m))) - (3*(e*cos(c + d*x))^(-3 - m)*(a + a*sin(c + d*x))^(1 + m))/(a*d*e*(1 - m)*(3 - m)) + (6*(e*cos(c + d*x))^(-3 - m)*(a + a*sin(c + d*x))^(2 + m))/(a^2*d*e*(3 - m)*(1 - m^2)) - (6*(e*cos(c + d*x))^(-3 - m)*(a + a*sin(c + d*x))^(3 + m))/(a^3*d*e*(9 - 10*m^2 + m^4))]
    @test_int [(e*cos(c + d*x))^(-3 - m)*(a + a*sin(c + d*x))^m, x, 3, -(((e*cos(c + d*x))^(-2 - m)*(a + a*sin(c + d*x))^m)/(d*e*(2 - m))) + (2*(e*cos(c + d*x))^(-2 - m)*(a + a*sin(c + d*x))^(1 + m))/(a*d*e*(2 - m)*m) - (2*(e*cos(c + d*x))^(-2 - m)*(a + a*sin(c + d*x))^(2 + m))/(a^2*d*e*m*(4 - m^2))]
    @test_int [(e*cos(c + d*x))^(-2 - m)*(a + a*sin(c + d*x))^m, x, 2, -(((e*cos(c + d*x))^(-1 - m)*(a + a*sin(c + d*x))^m)/(d*e*(1 - m))) + ((e*cos(c + d*x))^(-1 - m)*(a + a*sin(c + d*x))^(1 + m))/(a*d*e*(1 - m^2))]
    @test_int [(e*cos(c + d*x))^(-1 - m)*(a + a*sin(c + d*x))^m, x, 1, (a + a*sin(c + d*x))^m/((e*cos(c + d*x))^m*(d*e*m))]
    @test_int [(e*cos(c + d*x))^(0 - m)*(a + a*sin(c + d*x))^m, x, 3, -((2^(1/2 + m/2)*a*(e*cos(c + d*x))^(1 - m)*HypergeometricFunctions._₂F₁((1 - m)/2, (1 - m)/2, (3 - m)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1 - m)/2)*(a + a*sin(c + d*x))^(-1 + m))/(d*e*(1 - m)))]
    @test_int [(e*cos(c + d*x))^(1 - m)*(a + a*sin(c + d*x))^m, x, 3, (2^(1 - m/2)*(e*cos(c + d*x))^(2 - m)*HypergeometricFunctions._₂F₁(m/2, (2 + m)/2, (4 + m)/2, (1/2)*(1 + sin(c + d*x)))*(1 - sin(c + d*x))^(-1 + m/2)*(a + a*sin(c + d*x))^m)/(d*e*(2 + m))]
    @test_int [(e*cos(c + d*x))^(2 - m)*(a + a*sin(c + d*x))^m, x, 3, -((2^(3/2 + m/2)*a*(e*cos(c + d*x))^(3 - m)*HypergeometricFunctions._₂F₁((1/2)*(-1 - m), (3 - m)/2, (5 - m)/2, (1/2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^((1/2)*(-1 - m))*(a + a*sin(c + d*x))^(-1 + m))/(d*e*(3 - m)))]

    @test_int [(e*cos(c + d*x))^(5 - 2*m)*(a + a*sin(c + d*x))^m, x, 3, If(13>=8, -((8*a^3*(e*cos(c + d*x))^(6 - 2*m)*(a + a*sin(c + d*x))^(-3 + m))/(d*e*(5 - m)*(12 - 7*m + m^2))) - (4*a^2*(e*cos(c + d*x))^(6 - 2*m)*(a + a*sin(c + d*x))^(-2 + m))/(d*e*(20 - 9*m + m^2)) - (a*(e*cos(c + d*x))^(6 - 2*m)*(a + a*sin(c + d*x))^(-1 + m))/(d*e*(5 - m)), -((8*a^3*(e*cos(c + d*x))^(6 - 2*m)*(a + a*sin(c + d*x))^(-3 + m))/(d*e*(60 - 47*m + 12*m^2 - m^3))) - (4*a^2*(e*cos(c + d*x))^(6 - 2*m)*(a + a*sin(c + d*x))^(-2 + m))/(d*e*(4 - m)*(5 - m)) - (a*(e*cos(c + d*x))^(6 - 2*m)*(a + a*sin(c + d*x))^(-1 + m))/(d*e*(5 - m)))]
    @test_int [(e*cos(c + d*x))^(3 - 2*m)*(a + a*sin(c + d*x))^m, x, 2, If(13>=8, -((2*a^2*(e*cos(c + d*x))^(4 - 2*m)*(a + a*sin(c + d*x))^(-2 + m))/(d*e*(6 - 5*m + m^2))) - (a*(e*cos(c + d*x))^(4 - 2*m)*(a + a*sin(c + d*x))^(-1 + m))/(d*e*(3 - m)), -((2*a^2*(e*cos(c + d*x))^(4 - 2*m)*(a + a*sin(c + d*x))^(-2 + m))/(d*e*(2 - m)*(3 - m))) - (a*(e*cos(c + d*x))^(4 - 2*m)*(a + a*sin(c + d*x))^(-1 + m))/(d*e*(3 - m)))]
    @test_int [(e*cos(c + d*x))^(1 - 2*m)*(a + a*sin(c + d*x))^m, x, 1, -((a*(e*cos(c + d*x))^(2 - 2*m)*(a + a*sin(c + d*x))^(-1 + m))/(d*e*(1 - m)))]
    @test_int [(e*cos(c + d*x))^(-1 - 2*m)*(a + a*sin(c + d*x))^m, x, 3, (HypergeometricFunctions._₂F₁(1, -m, 1 - m, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^m)/((e*cos(c + d*x))^(2*m)*(2*d*e*m))]
    @test_int [(e*cos(c + d*x))^(-3 - 2*m)*(a + a*sin(c + d*x))^m, x, 3, (HypergeometricFunctions._₂F₁(2, -1 - m, -m, (1/2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(1 + m))/((e*cos(c + d*x))^(2*(1 + m))*(4*a*d*e*(1 + m)))]

    @test_int [(e*cos(c + d*x))^(4 - 2*m)*(a + a*sin(c + d*x))^m, x, 4, (2^(5/2 - m)*(e*cos(c + d*x))^(5 - 2*m)*HypergeometricFunctions._₂F₁(5/2, (1/2)*(-3 + 2*m), 7/2, (1/2)*(1 + sin(c + d*x)))*(1 - sin(c + d*x))^(-(5/2) + m)*(a + a*sin(c + d*x))^m)/(5*d*e)]
    @test_int [(e*cos(c + d*x))^(2 - 2*m)*(a + a*sin(c + d*x))^m, x, 4, (2^(3/2 - m)*(e*cos(c + d*x))^(3 - 2*m)*HypergeometricFunctions._₂F₁(3/2, (1/2)*(-1 + 2*m), 5/2, (1/2)*(1 + sin(c + d*x)))*(1 - sin(c + d*x))^(-(3/2) + m)*(a + a*sin(c + d*x))^m)/(3*d*e)]
    @test_int [(e*cos(c + d*x))^(0 - 2*m)*(a + a*sin(c + d*x))^m, x, 4, (2^(1/2 - m)*(e*cos(c + d*x))^(1 - 2*m)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + 2*m), 3/2, (1/2)*(1 + sin(c + d*x)))*(1 - sin(c + d*x))^(-(1/2) + m)*(a + a*sin(c + d*x))^m)/(d*e)]
    @test_int [(e*cos(c + d*x))^(-2 - 2*m)*(a + a*sin(c + d*x))^m, x, 4, -((2^(-(1/2) - m)*(e*cos(c + d*x))^(-1 - 2*m)*HypergeometricFunctions._₂F₁(-(1/2), (1/2)*(3 + 2*m), 1/2, (1/2)*(1 + sin(c + d*x)))*(1 - sin(c + d*x))^(1/2 + m)*(a + a*sin(c + d*x))^m)/(d*e))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*cos(e+f*x))^p*(a+b*sin(e+f*x))^m=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^p*(a+b*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x)), x, 4, -(b*cos(c + d*x)^6)/(6*d) + (a*sin(c + d*x))/d - (2*a*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x)), x, 3, -((b*cos(c + d*x)^4)/(4*d)) + (a*sin(c + d*x))/d - (a*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x)), x, 2, (a + b*sin(c + d*x))^2/(2*b*d), (a*sin(c + d*x))/d + (b*sin(c + d*x)^2)/(2*d)]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x)), x, 4, -((a + b)*log(1 - sin(c + d*x)))/(2*d) + ((a - b)*log(1 + sin(c + d*x)))/(2*d)]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x)), x, 3, (a*atanh(sin(c + d*x)))/(2*d) + (sec(c + d*x)^2*(b + a*sin(c + d*x)))/(2*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x)), x, 4, (3*a*atanh(sin(c + d*x)))/(8*d) + (sec(c + d*x)^4*(b + a*sin(c + d*x)))/(4*d) + (3*a*sec(c + d*x)*tan(c + d*x))/(8*d)]

    @test_int [cos(c + d*x)^4*(a + b*sin(c + d*x)), x, 4, (3*a*x)/8 - (b*cos(c + d*x)^5)/(5*d) + (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^2*(a + b*sin(c + d*x)), x, 3, (a*x)/2 - (b*cos(c + d*x)^3)/(3*d) + (a*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*(a + b*sin(c + d*x)), x, 3, (b*sec(c + d*x))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^4*(a + b*sin(c + d*x)), x, 3, (b*sec(c + d*x)^3)/(3*d) + (a*tan(c + d*x))/d + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^6*(a + b*sin(c + d*x)), x, 3, (b*sec(c + d*x)^5)/(5*d) + (a*tan(c + d*x))/d + (2*a*tan(c + d*x)^3)/(3*d) + (a*tan(c + d*x)^5)/(5*d)]

    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x))^2, x, 4, -((a*b*cos(c + d*x)^6)/(3*d)) + (a^2*sin(c + d*x))/d - ((2*a^2 - b^2)*sin(c + d*x)^3)/(3*d) + ((a^2 - 2*b^2)*sin(c + d*x)^5)/(5*d) + (b^2*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 3, -((a^2 - b^2)*(a + b*sin(c + d*x))^3)/(3*b^3*d) + (a*(a + b*sin(c + d*x))^4)/(2*b^3*d) - (a + b*sin(c + d*x))^5/(5*b^3*d)]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 2, (a + b*sin(c + d*x))^3/(3*b*d)]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x))^2, x, 6, -((a + b)^2*log(1 - sin(c + d*x)))/(2*d) + ((a - b)^2*log(1 + sin(c + d*x)))/(2*d) - (b^2*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x))^2, x, 3, ((a^2 - b^2)*atanh(sin(c + d*x)))/(2*d) + (sec(c + d*x)^2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(2*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x))^2, x, 4, ((3*a^2 - b^2)*atanh(sin(c + d*x)))/(8*d) + (sec(c + d*x)^4*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(4*d) + (sec(c + d*x)^2*(2*a*b + (3*a^2 - b^2)*sin(c + d*x)))/(8*d)]

    @test_int [cos(c + d*x)^6*(a + b*sin(c + d*x))^2, x, 6, (5/128)*(8*a^2 + b^2)*x - (9*a*b*cos(c + d*x)^7)/(56*d) + (5*(8*a^2 + b^2)*cos(c + d*x)*sin(c + d*x))/(128*d) + (5*(8*a^2 + b^2)*cos(c + d*x)^3*sin(c + d*x))/(192*d) + ((8*a^2 + b^2)*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (b*cos(c + d*x)^7*(a + b*sin(c + d*x)))/(8*d)]
    @test_int [cos(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 5, (1/16)*(6*a^2 + b^2)*x - (7*a*b*cos(c + d*x)^5)/(30*d) + ((6*a^2 + b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((6*a^2 + b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) - (b*cos(c + d*x)^5*(a + b*sin(c + d*x)))/(6*d)]
    @test_int [cos(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 4, (1/8)*(4*a^2 + b^2)*x - (5*a*b*cos(c + d*x)^3)/(12*d) + ((4*a^2 + b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) - (b*cos(c + d*x)^3*(a + b*sin(c + d*x)))/(4*d)]
    @test_int [sec(c + d*x)^2*(a + b*sin(c + d*x))^2, x, 3, (-b^2)*x + (a*b*cos(c + d*x))/d + (sec(c + d*x)*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^4*(a + b*sin(c + d*x))^2, x, 4, (a*b*sec(c + d*x))/(3*d) + (sec(c + d*x)^3*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(3*d) + ((2*a^2 - b^2)*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^6*(a + b*sin(c + d*x))^2, x, 4, (a*b*sec(c + d*x)^3)/(5*d) + (sec(c + d*x)^5*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(5*d) + ((4*a^2 - b^2)*tan(c + d*x))/(5*d) + ((4*a^2 - b^2)*tan(c + d*x)^3)/(15*d)]
    @test_int [sec(c + d*x)^8*(a + b*sin(c + d*x))^2, x, 4, (a*b*sec(c + d*x)^5)/(7*d) + (sec(c + d*x)^7*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(7*d) + ((6*a^2 - b^2)*tan(c + d*x))/(7*d) + (2*(6*a^2 - b^2)*tan(c + d*x)^3)/(21*d) + ((6*a^2 - b^2)*tan(c + d*x)^5)/(35*d)]

    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x))^3, x, 3, ((a^2 - b^2)^2*(a + b*sin(c + d*x))^4)/(4*b^5*d) - (4*a*(a^2 - b^2)*(a + b*sin(c + d*x))^5)/(5*b^5*d) + ((3*a^2 - b^2)*(a + b*sin(c + d*x))^6)/(3*b^5*d) - (4*a*(a + b*sin(c + d*x))^7)/(7*b^5*d) + (a + b*sin(c + d*x))^8/(8*b^5*d)]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x))^3, x, 3, -((a^2 - b^2)*(a + b*sin(c + d*x))^4)/(4*b^3*d) + (2*a*(a + b*sin(c + d*x))^5)/(5*b^3*d) - (a + b*sin(c + d*x))^6/(6*b^3*d)]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 2, (a + b*sin(c + d*x))^4/(4*b*d)]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x))^3, x, 6, -((a + b)^3*log(1 - sin(c + d*x)))/(2*d) + ((a - b)^3*log(1 + sin(c + d*x)))/(2*d) - (3*a*b^2*sin(c + d*x))/d - (b^3*sin(c + d*x)^2)/(2*d)]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x))^3, x, 6, -((a - 2*b)*(a + b)^2*log(1 - sin(c + d*x)))/(4*d) + ((a - b)^2*(a + 2*b)*log(1 + sin(c + d*x)))/(4*d) + (a*b^2*sin(c + d*x))/(2*d) + (sec(c + d*x)^2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(2*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x))^3, x, 4, (3*a*(a^2 - b^2)*atanh(sin(c + d*x)))/(8*d) + (3*a*sec(c + d*x)^2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(8*d) + (sec(c + d*x)^3*(a + b*sin(c + d*x))^3*tan(c + d*x))/(4*d)]

    @test_int [cos(c + d*x)^4*(a + b*sin(c + d*x))^3, x, 6, (3/16)*a*(2*a^2 + b^2)*x - (b*(17*a^2 + 4*b^2)*cos(c + d*x)^5)/(70*d) + (3*a*(2*a^2 + b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*(2*a^2 + b^2)*cos(c + d*x)^3*sin(c + d*x))/(8*d) - (3*a*b*cos(c + d*x)^5*(a + b*sin(c + d*x)))/(14*d) - (b*cos(c + d*x)^5*(a + b*sin(c + d*x))^2)/(7*d)]
    @test_int [cos(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 5, (1/8)*a*(4*a^2 + 3*b^2)*x - (b*(27*a^2 + 8*b^2)*cos(c + d*x)^3)/(60*d) + (a*(4*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) - (7*a*b*cos(c + d*x)^3*(a + b*sin(c + d*x)))/(20*d) - (b*cos(c + d*x)^3*(a + b*sin(c + d*x))^2)/(5*d)]
    @test_int [sec(c + d*x)^2*(a + b*sin(c + d*x))^3, x, 2, -3*a*b^2*x + (2*b*(a^2 + b^2)*cos(c + d*x))/d + (a*b^2*cos(c + d*x)*sin(c + d*x))/d + (sec(c + d*x)*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/d]
    @test_int [sec(c + d*x)^4*(a + b*sin(c + d*x))^3, x, 5, (2*b*(a^2 - b^2)*sec(c + d*x))/(3*d) + (sec(c + d*x)^3*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(3*d) + (2*a*(a^2 - b^2)*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^6*(a + b*sin(c + d*x))^3, x, 5, (2*b*(2*a^2 - b^2)*sec(c + d*x))/(15*d) + (sec(c + d*x)^5*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(5*d) + (2*sec(c + d*x)^3*(a + b*sin(c + d*x))*(a*b + (2*a^2 - b^2)*sin(c + d*x)))/(15*d) + (2*a*(4*a^2 - 3*b^2)*tan(c + d*x))/(15*d)]
    @test_int [sec(c + d*x)^8*(a + b*sin(c + d*x))^3, x, 5, (2*b*(3*a^2 - b^2)*sec(c + d*x)^3)/(35*d) + (sec(c + d*x)^7*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(7*d) + (2*sec(c + d*x)^5*(a + b*sin(c + d*x))*(2*a*b + (3*a^2 - b^2)*sin(c + d*x)))/(35*d) + (12*a*(2*a^2 - b^2)*tan(c + d*x))/(35*d) + (4*a*(2*a^2 - b^2)*tan(c + d*x)^3)/(35*d)]
    @test_int [sec(c + d*x)^10*(a + b*sin(c + d*x))^3, x, 5, (2*b*(4*a^2 - b^2)*sec(c + d*x)^5)/(63*d) + (sec(c + d*x)^9*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(9*d) + (2*sec(c + d*x)^7*(a + b*sin(c + d*x))*(3*a*b + (4*a^2 - b^2)*sin(c + d*x)))/(63*d) + (2*a*(8*a^2 - 3*b^2)*tan(c + d*x))/(21*d) + (4*a*(8*a^2 - 3*b^2)*tan(c + d*x)^3)/(63*d) + (2*a*(8*a^2 - 3*b^2)*tan(c + d*x)^5)/(105*d)]

    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x))^8, x, 3, ((a^2 - b^2)^2*(a + b*sin(c + d*x))^9)/(9*b^5*d) - (2*a*(a^2 - b^2)*(a + b*sin(c + d*x))^10)/(5*b^5*d) + (2*(3*a^2 - b^2)*(a + b*sin(c + d*x))^11)/(11*b^5*d) - (a*(a + b*sin(c + d*x))^12)/(3*b^5*d) + (a + b*sin(c + d*x))^13/(13*b^5*d)]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x))^8, x, 3, -((a^2 - b^2)*(a + b*sin(c + d*x))^9)/(9*b^3*d) + (a*(a + b*sin(c + d*x))^10)/(5*b^3*d) - (a + b*sin(c + d*x))^11/(11*b^3*d)]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x))^8, x, 2, (a + b*sin(c + d*x))^9/(9*b*d)]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x))^8, x, 6, -((a + b)^8*log(1 - sin(c + d*x)))/(2*d) + ((a - b)^8*log(1 + sin(c + d*x)))/(2*d) - (b^2*(28*a^6 + 70*a^4*b^2 + 28*a^2*b^4 + b^6)*sin(c + d*x))/d - (4*a*b^3*(7*a^4 + 7*a^2*b^2 + b^4)*sin(c + d*x)^2)/d - (b^4*(70*a^4 + 28*a^2*b^2 + b^4)*sin(c + d*x)^3)/(3*d) - (2*a*b^5*(7*a^2 + b^2)*sin(c + d*x)^4)/d - (b^6*(28*a^2 + b^2)*sin(c + d*x)^5)/(5*d) - (4*a*b^7*sin(c + d*x)^6)/(3*d) - (b^8*sin(c + d*x)^7)/(7*d)]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x))^8, x, 7, -((a - 7*b)*(a + b)^7*log(1 - sin(c + d*x)))/(4*d) + ((a - b)^7*(a + 7*b)*log(1 + sin(c + d*x)))/(4*d) + (7*b^2*(3*a^6 + 30*a^4*b^2 + 20*a^2*b^4 + b^6)*sin(c + d*x))/(2*d) + (a*b^3*(35*a^4 + 112*a^2*b^2 + 24*b^4)*sin(c + d*x)^2)/(2*d) + (7*b^4*(15*a^4 + 20*a^2*b^2 + b^4)*sin(c + d*x)^3)/(6*d) + (3*a*b^5*(7*a^2 + 4*b^2)*sin(c + d*x)^4)/(2*d) + (7*b^6*(5*a^2 + b^2)*sin(c + d*x)^5)/(10*d) + (a*b^7*sin(c + d*x)^6)/(2*d) + (sec(c + d*x)^2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^7)/(2*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x))^8, x, 8, -(((a + b)^6*(3*a^2 - 18*a*b + 35*b^2)*log(1 - sin(c + d*x)))/(16*d)) + ((a - b)^6*(3*a^2 + 18*a*b + 35*b^2)*log(1 + sin(c + d*x)))/(16*d) + (5*b^2*(6*a^6 - 35*a^4*b^2 - 84*a^2*b^4 - 7*b^6)*sin(c + d*x))/(8*d) + (a*b^3*(15*a^4 - 77*a^2*b^2 - 48*b^4)*sin(c + d*x)^2)/(4*d) + (5*b^4*(9*a^4 - 42*a^2*b^2 - 7*b^4)*sin(c + d*x)^3)/(24*d) - (a*(13 - (3*a^2)/b^2)*b^7*sin(c + d*x)^4)/(8*d) + (sec(c + d*x)^4*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^7)/(4*d) - (sec(c + d*x)^2*(a + b*sin(c + d*x))^5*(b*(a^2 + 7*b^2) - a*(3*a^2 - 11*b^2)*sin(c + d*x)))/(8*d)]

    @test_int [cos(c + d*x)^2*(a + b*sin(c + d*x))^8, x, 10, (1/256)*(128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*x - (11*a*b*(1792*a^6 + 10536*a^4*b^2 + 9588*a^2*b^4 + 1289*b^6)*cos(c + d*x)^3)/(40320*d) + ((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*cos(c + d*x)*sin(c + d*x))/(256*d) - (b*(6272*a^6 + 28088*a^4*b^2 + 15956*a^2*b^4 + 735*b^6)*cos(c + d*x)^3*(a + b*sin(c + d*x)))/(13440*d) - (13*a*b*(112*a^4 + 348*a^2*b^2 + 101*b^4)*cos(c + d*x)^3*(a + b*sin(c + d*x))^2)/(3360*d) - (b*(784*a^4 + 1500*a^2*b^2 + 147*b^4)*cos(c + d*x)^3*(a + b*sin(c + d*x))^3)/(2016*d) - (a*b*(112*a^2 + 109*b^2)*cos(c + d*x)^3*(a + b*sin(c + d*x))^4)/(336*d) - (b*(64*a^2 + 21*b^2)*cos(c + d*x)^3*(a + b*sin(c + d*x))^5)/(240*d) - (17*a*b*cos(c + d*x)^3*(a + b*sin(c + d*x))^6)/(90*d) - (b*cos(c + d*x)^3*(a + b*sin(c + d*x))^7)/(10*d)]
    @test_int [sec(c + d*x)^2*(a + b*sin(c + d*x))^8, x, 7, (-(7/16))*b^2*(64*a^6 + 240*a^4*b^2 + 120*a^2*b^4 + 5*b^6)*x + (a*b*(40*a^6 + 1664*a^4*b^2 + 2789*a^2*b^4 + 512*b^6)*cos(c + d*x))/(20*d) + (b^2*(80*a^6 + 2248*a^4*b^2 + 2502*a^2*b^4 + 175*b^6)*cos(c + d*x)*sin(c + d*x))/(80*d) + (a*b*(40*a^4 + 624*a^2*b^2 + 337*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(40*d) + (b*(120*a^4 + 992*a^2*b^2 + 175*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^3)/(120*d) + (a*b*(30*a^2 + 113*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^4)/(30*d) + (b*(6*a^2 + 7*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^5)/(6*d) + (a*b*cos(c + d*x)*(a + b*sin(c + d*x))^6)/d + (sec(c + d*x)*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^7)/d]
    @test_int [sec(c + d*x)^4*(a + b*sin(c + d*x))^8, x, 7, (35/8)*b^4*(16*a^4 + 16*a^2*b^2 + b^4)*x + (a*b*(8*a^6 - 104*a^4*b^2 - 803*a^2*b^4 - 256*b^6)*cos(c + d*x))/(6*d) + (b^2*(16*a^6 - 200*a^4*b^2 - 866*a^2*b^4 - 105*b^6)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*b*(8*a^4 - 88*a^2*b^2 - 151*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(12*d) + (b*(8*a^4 - 72*a^2*b^2 - 35*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^3)/(12*d) + (a*b*(2*a^2 - 13*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^4)/(3*d) + (b*(2*a^2 - 7*b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^5)/(3*d) + (sec(c + d*x)^3*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^7)/(3*d) - (sec(c + d*x)*(a + b*sin(c + d*x))^6*(5*a*b - (2*a^2 - 7*b^2)*sin(c + d*x)))/(3*d)]
    @test_int [sec(c + d*x)^6*(a + b*sin(c + d*x))^8, x, 7, (-(7/2))*b^6*(8*a^2 + b^2)*x + (2*a*b*(8*a^6 - 48*a^4*b^2 + 163*a^2*b^4 + 192*b^6)*cos(c + d*x))/(15*d) + (b^2*(16*a^6 - 88*a^4*b^2 + 282*a^2*b^4 + 105*b^6)*cos(c + d*x)*sin(c + d*x))/(30*d) + (a*b*(8*a^4 - 32*a^2*b^2 + 87*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(15*d) + (b*(8*a^4 - 16*a^2*b^2 + 35*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^3)/(15*d) + (4*a*b*(2*a^2 + b^2)*cos(c + d*x)*(a + b*sin(c + d*x))^4)/(15*d) + (sec(c + d*x)^5*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^7)/(5*d) - (sec(c + d*x)^3*(a + b*sin(c + d*x))^6*(3*a*b - (4*a^2 - 7*b^2)*sin(c + d*x)))/(15*d) - (4*sec(c + d*x)*(a + b*sin(c + d*x))^5*(b*(4*a^2 - 7*b^2) - a*(2*a^2 + b^2)*sin(c + d*x)))/(15*d)]
    @test_int [sec(c + d*x)^8*(a + b*sin(c + d*x))^8, x, 7, b^8*x + (4*a*b*(24*a^6 - 88*a^4*b^2 + 125*a^2*b^4 - 96*b^6)*cos(c + d*x))/(105*d) + (b^2*(48*a^6 - 152*a^4*b^2 + 174*a^2*b^4 - 105*b^6)*cos(c + d*x)*sin(c + d*x))/(105*d) + (2*a*b*(24*a^4 - 40*a^2*b^2 + 9*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^2)/(105*d) + (2*b*(24*a^4 + 8*a^2*b^2 - 35*b^4)*cos(c + d*x)*(a + b*sin(c + d*x))^3)/(105*d) + (sec(c + d*x)^7*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^7)/(7*d) - (2*sec(c + d*x)^3*(a + b*sin(c + d*x))^5*(b*(6*a^2 - 7*b^2) - a*(12*a^2 - 11*b^2)*sin(c + d*x)))/(105*d) - (sec(c + d*x)^5*(a + b*sin(c + d*x))^6*(a*b - (6*a^2 - 7*b^2)*sin(c + d*x)))/(35*d) - (2*sec(c + d*x)*(a + b*sin(c + d*x))^4*(3*a*b*(12*a^2 - 11*b^2) - (24*a^4 + 8*a^2*b^2 - 35*b^4)*sin(c + d*x)))/(105*d)]
    @test_int [sec(c + d*x)^10*(a + b*sin(c + d*x))^8, x, 10, (128*a*b*(a^2 - b^2)^3*sec(c + d*x))/(315*d) + (64*a*(a^2 - b^2)^2*sec(c + d*x)^3*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(315*d) + (16*a*(a^2 - b^2)*sec(c + d*x)^5*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^4)/(105*d) + (sec(c + d*x)^9*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^7)/(9*d) + (sec(c + d*x)^7*(a + b*sin(c + d*x))^6*(a*b + (8*a^2 - 7*b^2)*sin(c + d*x)))/(63*d) + (128*a^2*(a^2 - b^2)^3*tan(c + d*x))/(315*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^5/(a + b*sin(c + d*x)), x, 3, ((a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(b^5*d) - (a*(a^2 - 2*b^2)*sin(c + d*x))/(b^4*d) + ((a^2 - 2*b^2)*sin(c + d*x)^2)/(2*b^3*d) - (a*sin(c + d*x)^3)/(3*b^2*d) + sin(c + d*x)^4/(4*b*d)]
    @test_int [cos(c + d*x)^3/(a + b*sin(c + d*x)), x, 3, -(((a^2 - b^2)*log(a + b*sin(c + d*x)))/(b^3*d)) + (a*sin(c + d*x))/(b^2*d) - sin(c + d*x)^2/(2*b*d)]
    @test_int [cos(c + d*x)^1/(a + b*sin(c + d*x)), x, 2, log(a + b*sin(c + d*x))/(b*d)]
    @test_int [sec(c + d*x)^1/(a + b*sin(c + d*x)), x, 6, -log(1 - sin(c + d*x))/(2*(a + b)*d) + log(1 + sin(c + d*x))/(2*(a - b)*d) - (b*log(a + b*sin(c + d*x)))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^3/(a + b*sin(c + d*x)), x, 4, -((a + 2*b)*log(1 - sin(c + d*x)))/(4*(a + b)^2*d) + ((a - 2*b)*log(1 + sin(c + d*x)))/(4*(a - b)^2*d) + (b^3*log(a + b*sin(c + d*x)))/((a^2 - b^2)^2*d) - (sec(c + d*x)^2*(b - a*sin(c + d*x)))/(2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^5/(a + b*sin(c + d*x)), x, 5, -((3*a^2 + 9*a*b + 8*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^3*d) + ((3*a^2 - 9*a*b + 8*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^3*d) - (b^5*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d) + (sec(c + d*x)^2*(4*b^3 + a*(3*a^2 - 7*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]

    @test_int [cos(c + d*x)^6/(a + b*sin(c + d*x)), x, 7, (a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) - (2*(a^2 - b^2)^(5/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^6*d) + cos(c + d*x)^5/(5*b*d) - (cos(c + d*x)^3*(4*(a^2 - b^2) - 3*a*b*sin(c + d*x)))/(12*b^3*d) + (cos(c + d*x)*(8*(a^2 - b^2)^2 - a*b*(4*a^2 - 7*b^2)*sin(c + d*x)))/(8*b^5*d)]
    @test_int [cos(c + d*x)^4/(a + b*sin(c + d*x)), x, 6, -((a*(2*a^2 - 3*b^2)*x)/(2*b^4)) + (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*d) + cos(c + d*x)^3/(3*b*d) - (cos(c + d*x)*(2*(a^2 - b^2) - a*b*sin(c + d*x)))/(2*b^3*d)]
    @test_int [cos(c + d*x)^2/(a + b*sin(c + d*x)), x, 5, (a*x)/b^2 - (2*sqrt(a^2 - b^2)*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^2*d) + cos(c + d*x)/(b*d)]
    @test_int [sec(c + d*x)^2/(a + b*sin(c + d*x)), x, 5, -((2*b^2*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d)) - (sec(c + d*x)*(b - a*sin(c + d*x)))/((a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^4/(a + b*sin(c + d*x)), x, 6, (2*b^4*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - (sec(c + d*x)^3*(b - a*sin(c + d*x)))/(3*(a^2 - b^2)*d) + (sec(c + d*x)*(3*b^3 + a*(2*a^2 - 5*b^2)*sin(c + d*x)))/(3*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^6/(a + b*sin(c + d*x)), x, 7, -((2*b^6*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) - (sec(c + d*x)^5*(b - a*sin(c + d*x)))/(5*(a^2 - b^2)*d) + (sec(c + d*x)^3*(5*b^3 + a*(4*a^2 - 9*b^2)*sin(c + d*x)))/(15*(a^2 - b^2)^2*d) - (sec(c + d*x)*(15*b^5 - a*(8*a^4 - 26*a^2*b^2 + 33*b^4)*sin(c + d*x)))/(15*(a^2 - b^2)^3*d)]

    @test_int [cos(c + d*x)^7/(a + b*sin(c + d*x))^2, x, 3, (6*a*(a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(b^7*d) - ((5*a^4 - 9*a^2*b^2 + 3*b^4)*sin(c + d*x))/(b^6*d) + (a*(2*a^2 - 3*b^2)*sin(c + d*x)^2)/(b^5*d) - ((a^2 - b^2)*sin(c + d*x)^3)/(b^4*d) + (a*sin(c + d*x)^4)/(2*b^3*d) - sin(c + d*x)^5/(5*b^2*d) + (a^2 - b^2)^3/(b^7*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5/(a + b*sin(c + d*x))^2, x, 3, (-4*a*(a^2 - b^2)*log(a + b*sin(c + d*x)))/(b^5*d) + ((3*a^2 - 2*b^2)*sin(c + d*x))/(b^4*d) - (a*sin(c + d*x)^2)/(b^3*d) + sin(c + d*x)^3/(3*b^2*d) - (a^2 - b^2)^2/(b^5*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 3, (2*a*log(a + b*sin(c + d*x)))/(b^3*d) - sin(c + d*x)/(b^2*d) + (a^2 - b^2)/(b^3*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 2, -(1/(b*d*(a + b*sin(c + d*x))))]
    @test_int [sec(c + d*x)^1/(a + b*sin(c + d*x))^2, x, 4, -log(1 - sin(c + d*x))/(2*(a + b)^2*d) + log(1 + sin(c + d*x))/(2*(a - b)^2*d) - (2*a*b*log(a + b*sin(c + d*x)))/((a^2 - b^2)^2*d) + b/((a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*sin(c + d*x))^2, x, 4, -((a + 3*b)*log(1 - sin(c + d*x)))/(4*(a + b)^3*d) + ((a - 3*b)*log(1 + sin(c + d*x)))/(4*(a - b)^3*d) + (4*a*b^3*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) - (b*(a^2 + 3*b^2))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) - (sec(c + d*x)^2*(b - a*sin(c + d*x)))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5/(a + b*sin(c + d*x))^2, x, 5, (-3*(a^2 + 4*a*b + 5*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^4*d) + (3*(a^2 - 4*a*b + 5*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^4*d) - (6*a*b^5*log(a + b*sin(c + d*x)))/((a^2 - b^2)^4*d) - (3*b*(a^4 - 4*a^2*b^2 - 5*b^4))/(8*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d*(a + b*sin(c + d*x))) + (sec(c + d*x)^2*(b*(a^2 + 5*b^2) + 3*a*(a^2 - 3*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]

    @test_int [cos(c + d*x)^6/(a + b*sin(c + d*x))^2, x, 7, -((5*(8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^6)) + (10*a*(a^2 - b^2)^(3/2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^6*d) + (5*cos(c + d*x)^3*(4*a - 3*b*sin(c + d*x)))/(12*b^3*d) - cos(c + d*x)^5/(b*d*(a + b*sin(c + d*x))) - (5*cos(c + d*x)*(8*a*(a^2 - b^2) - b*(4*a^2 - 3*b^2)*sin(c + d*x)))/(8*b^5*d)]
    @test_int [cos(c + d*x)^4/(a + b*sin(c + d*x))^2, x, 6, (3*(2*a^2 - b^2)*x)/(2*b^4) - (6*a*sqrt(a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*d) + (3*cos(c + d*x)*(2*a - b*sin(c + d*x)))/(2*b^3*d) - cos(c + d*x)^3/(b*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 5, -(x/b^2) + (2*a*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*d) - cos(c + d*x)/(b*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sin(c + d*x))^2, x, 6, -((6*a*b^2*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d)) + (b*sec(c + d*x))/((a^2 - b^2)*d*(a + b*sin(c + d*x))) - (sec(c + d*x)*(3*a*b - (a^2 + 2*b^2)*sin(c + d*x)))/((a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^4/(a + b*sin(c + d*x))^2, x, 7, (10*a*b^4*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) + (b*sec(c + d*x)^3)/((a^2 - b^2)*d*(a + b*sin(c + d*x))) - (sec(c + d*x)^3*(5*a*b - (a^2 + 4*b^2)*sin(c + d*x)))/(3*(a^2 - b^2)^2*d) + (sec(c + d*x)*(15*a*b^3 + (2*a^4 - 9*a^2*b^2 - 8*b^4)*sin(c + d*x)))/(3*(a^2 - b^2)^3*d)]

    @test_int [cos(c + d*x)^7/(a + b*sin(c + d*x))^3, x, 3, -((3*(5*a^4 - 6*a^2*b^2 + b^4)*log(a + b*sin(c + d*x)))/(b^7*d)) + (a*(10*a^2 - 9*b^2)*sin(c + d*x))/(b^6*d) - (3*(2*a^2 - b^2)*sin(c + d*x)^2)/(2*b^5*d) + (a*sin(c + d*x)^3)/(b^4*d) - sin(c + d*x)^4/(4*b^3*d) + (a^2 - b^2)^3/(2*b^7*d*(a + b*sin(c + d*x))^2) - (6*a*(a^2 - b^2)^2)/(b^7*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5/(a + b*sin(c + d*x))^3, x, 3, (2*(3*a^2 - b^2)*log(a + b*sin(c + d*x)))/(b^5*d) - (3*a*sin(c + d*x))/(b^4*d) + sin(c + d*x)^2/(2*b^3*d) - (a^2 - b^2)^2/(2*b^5*d*(a + b*sin(c + d*x))^2) + (4*a*(a^2 - b^2))/(b^5*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 3, -(log(a + b*sin(c + d*x))/(b^3*d)) + (a^2 - b^2)/(2*b^3*d*(a + b*sin(c + d*x))^2) - (2*a)/(b^3*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 2, -1/(2*b*d*(a + b*sin(c + d*x))^2)]
    @test_int [sec(c + d*x)^1/(a + b*sin(c + d*x))^3, x, 4, -log(1 - sin(c + d*x))/(2*(a + b)^3*d) + log(1 + sin(c + d*x))/(2*(a - b)^3*d) - (b*(3*a^2 + b^2)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^3*d) + b/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) + (2*a*b)/((a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*sin(c + d*x))^3, x, 4, -((a + 4*b)*log(1 - sin(c + d*x)))/(4*(a + b)^4*d) + ((a - 4*b)*log(1 + sin(c + d*x)))/(4*(a - b)^4*d) + (2*b^3*(5*a^2 + b^2)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^4*d) - (b*(a^2 + 2*b^2))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2) - (sec(c + d*x)^2*(b - a*sin(c + d*x)))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) - (a*b*(a^2 + 11*b^2))/(2*(a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5/(a + b*sin(c + d*x))^3, x, 5, (-3*(a^2 + 5*a*b + 8*b^2)*log(1 - sin(c + d*x)))/(16*(a + b)^5*d) + (3*(a^2 - 5*a*b + 8*b^2)*log(1 + sin(c + d*x)))/(16*(a - b)^5*d) - (3*b^5*(7*a^2 + b^2)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^5*d) - (3*b*(a^4 - 5*a^2*b^2 - 4*b^4))/(8*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^2) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) - (3*a*b*(a^4 - 6*a^2*b^2 - 27*b^4))/(8*(a^2 - b^2)^4*d*(a + b*sin(c + d*x))) + (sec(c + d*x)^2*(2*b*(a^2 + 3*b^2) + a*(3*a^2 - 11*b^2)*sin(c + d*x)))/(8*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^2)]

    @test_int [cos(c + d*x)^6/(a + b*sin(c + d*x))^3, x, 7, (5*a*(4*a^2 - 3*b^2)*x)/(2*b^6) - (5*(4*a^4 - 5*a^2*b^2 + b^4)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^6*sqrt(a^2 - b^2)*d) - cos(c + d*x)^5/(2*b*d*(a + b*sin(c + d*x))^2) - (5*cos(c + d*x)^3*(4*a + b*sin(c + d*x)))/(6*b^3*d*(a + b*sin(c + d*x))) + (5*cos(c + d*x)*(4*a^2 - b^2 - 2*a*b*sin(c + d*x)))/(2*b^5*d)]
    @test_int [cos(c + d*x)^4/(a + b*sin(c + d*x))^3, x, 6, -((3*a*x)/b^4) + (3*(2*a^2 - b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)*d) - cos(c + d*x)^3/(2*b*d*(a + b*sin(c + d*x))^2) - (3*cos(c + d*x)*(2*a + b*sin(c + d*x)))/(2*b^3*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 6, atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2))/((a^2 - b^2)^(3/2)*d) - cos(c + d*x)/(2*b*d*(a + b*sin(c + d*x))^2) + (a*cos(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sin(c + d*x))^3, x, 7, -((3*b^2*(4*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) + (b*sec(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) + (5*a*b*sec(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) - (sec(c + d*x)*(3*b*(4*a^2 + b^2) - a*(2*a^2 + 13*b^2)*sin(c + d*x)))/(2*(a^2 - b^2)^3*d)]
    @test_int [sec(c + d*x)^4/(a + b*sin(c + d*x))^3, x, 8, (5*b^4*(6*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(9/2)*d) + (b*sec(c + d*x)^3)/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) + (7*a*b*sec(c + d*x)^3)/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) - (sec(c + d*x)^3*(5*b*(6*a^2 + b^2) - a*(2*a^2 + 33*b^2)*sin(c + d*x)))/(6*(a^2 - b^2)^3*d) + (sec(c + d*x)*(15*b^3*(6*a^2 + b^2) + a*(4*a^4 - 28*a^2*b^2 - 81*b^4)*sin(c + d*x)))/(6*(a^2 - b^2)^4*d)]

    @test_int [cos(c + d*x)^7/(a + b*sin(c + d*x))^8, x, 3, (a^2 - b^2)^3/(7*b^7*d*(a + b*sin(c + d*x))^7) - (a*(a^2 - b^2)^2)/(b^7*d*(a + b*sin(c + d*x))^6) + (3*(5*a^4 - 6*a^2*b^2 + b^4))/(5*b^7*d*(a + b*sin(c + d*x))^5) - (a*(5*a^2 - 3*b^2))/(b^7*d*(a + b*sin(c + d*x))^4) + (5*a^2 - b^2)/(b^7*d*(a + b*sin(c + d*x))^3) - (3*a)/(b^7*d*(a + b*sin(c + d*x))^2) + 1/(b^7*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^5/(a + b*sin(c + d*x))^8, x, 3, -(a^2 - b^2)^2/(7*b^5*d*(a + b*sin(c + d*x))^7) + (2*a*(a^2 - b^2))/(3*b^5*d*(a + b*sin(c + d*x))^6) - (2*(3*a^2 - b^2))/(5*b^5*d*(a + b*sin(c + d*x))^5) + a/(b^5*d*(a + b*sin(c + d*x))^4) - 1/(3*b^5*d*(a + b*sin(c + d*x))^3)]
    @test_int [cos(c + d*x)^3/(a + b*sin(c + d*x))^8, x, 3, (a^2 - b^2)/(7*b^3*d*(a + b*sin(c + d*x))^7) - a/(3*b^3*d*(a + b*sin(c + d*x))^6) + 1/(5*b^3*d*(a + b*sin(c + d*x))^5)]
    @test_int [cos(c + d*x)^1/(a + b*sin(c + d*x))^8, x, 2, -1/(7*b*d*(a + b*sin(c + d*x))^7)]
    @test_int [sec(c + d*x)^1/(a + b*sin(c + d*x))^8, x, 4, If(13>=8, -log(1 - sin(c + d*x))/(2*(a + b)^8*d) + log(1 + sin(c + d*x))/(2*(a - b)^8*d) - (8*a*b*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^8*d) + b/(7*(a^2 - b^2)*d*(a + b*sin(c + d*x))^7) + (a*b)/(3*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^6) + (b*(3*a^2 + b^2))/(5*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^5) + (a*b*(a^2 + b^2))/((a^2 - b^2)^4*d*(a + b*sin(c + d*x))^4) + (b*(5*a^4 + 10*a^2*b^2 + b^4))/(3*(a^2 - b^2)^5*d*(a + b*sin(c + d*x))^3) + (a*b*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a^2 - b^2)^6*d*(a + b*sin(c + d*x))^2) + (b*(7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6))/((a^2 - b^2)^7*d*(a + b*sin(c + d*x))), -(log(1 - sin(c + d*x))/(2*(a + b)^8*d)) + log(1 + sin(c + d*x))/(2*(a - b)^8*d) - (8*a*b*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^8*d) + b/(7*(a^2 - b^2)*d*(a + b*sin(c + d*x))^7) + (a*b)/(3*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^6) + (b*(3*a^2 + b^2))/(5*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^5) + (a*b*(a^2 + b^2))/((a^2 - b^2)^4*d*(a + b*sin(c + d*x))^4) + (b*(5*a^4 + 10*a^2*b^2 + b^4))/(3*(a^2 - b^2)^5*d*(a + b*sin(c + d*x))^3) + (a*b*(3*a^4 + 10*a^2*b^2 + 3*b^4))/((a^2 - b^2)^6*d*(a + b*sin(c + d*x))^2) + (b*(7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6))/((a^2 - b^2)^7*d*(a + b*sin(c + d*x))))]
    @test_int [sec(c + d*x)^3/(a + b*sin(c + d*x))^8, x, 4, -((a + 9*b)*log(1 - sin(c + d*x)))/(4*(a + b)^9*d) + ((a - 9*b)*log(1 + sin(c + d*x)))/(4*(a - b)^9*d) + (8*a*b^3*(15*a^6 + 63*a^4*b^2 + 45*a^2*b^4 + 5*b^6)*log(a + b*sin(c + d*x)))/((a^2 - b^2)^9*d) - (b*(7*a^2 + 9*b^2))/(14*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^7) - (sec(c + d*x)^2*(b - a*sin(c + d*x)))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^7) - (a*b*(3*a^2 + 13*b^2))/(6*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^6) - (b*(5*a^4 + 50*a^2*b^2 + 9*b^4))/(10*(a^2 - b^2)^4*d*(a + b*sin(c + d*x))^5) - (a*b*(a^4 + 20*a^2*b^2 + 11*b^4))/(2*(a^2 - b^2)^5*d*(a + b*sin(c + d*x))^4) - (b*(3*a^6 + 115*a^4*b^2 + 129*a^2*b^4 + 9*b^6))/(6*(a^2 - b^2)^6*d*(a + b*sin(c + d*x))^3) - (a*b*(a^6 + 77*a^4*b^2 + 147*a^2*b^4 + 31*b^6))/(2*(a^2 - b^2)^7*d*(a + b*sin(c + d*x))^2) - (b*(a^8 + 196*a^6*b^2 + 574*a^4*b^4 + 244*a^2*b^6 + 9*b^8))/(2*(a^2 - b^2)^8*d*(a + b*sin(c + d*x)))]

    @test_int [cos(c + d*x)^8/(a + b*sin(c + d*x))^8, x, 11, x/b^8 - (a*(16*a^6 - 56*a^4*b^2 + 70*a^2*b^4 - 35*b^6)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(8*b^8*(a^2 - b^2)^(7/2)*d) - cos(c + d*x)^7/(7*b*d*(a + b*sin(c + d*x))^7) + (a*cos(c + d*x)^7)/(6*b*(a^2 - b^2)*d*(a + b*sin(c + d*x))^6) - (a*(6*a^2 - 11*b^2)*cos(c + d*x)^5)/(24*b^3*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^4) + (a*(8*a^4 - 22*a^2*b^2 + 19*b^4)*cos(c + d*x)^3)/(16*b^5*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^2) + (cos(c + d*x)^5*(6*(a^2 - b^2) + 5*a*b*sin(c + d*x)))/(30*b^3*(a^2 - b^2)*d*(a + b*sin(c + d*x))^5) - (cos(c + d*x)^3*(8*(a^2 - b^2)^2 + a*b*(6*a^2 - 11*b^2)*sin(c + d*x)))/(24*b^5*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^3) + (cos(c + d*x)*(16*(a^2 - b^2)^3 + a*b*(8*a^4 - 22*a^2*b^2 + 19*b^4)*sin(c + d*x)))/(16*b^7*(a^2 - b^2)^3*d*(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^6/(a + b*sin(c + d*x))^8, x, 11, (5*a*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(8*(a^2 - b^2)^(9/2)*d) - cos(c + d*x)^5/(7*b*d*(a + b*sin(c + d*x))^7) + (a*(4*a^2 - b^2)*cos(c + d*x))/(168*b^5*(a^2 - b^2)*d*(a + b*sin(c + d*x))^4) + ((4*a^4 - 9*a^2*b^2 + 12*b^4)*cos(c + d*x))/(168*b^5*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^3) + (a*(8*a^4 - 30*a^2*b^2 + 57*b^4)*cos(c + d*x))/(336*b^5*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^2) + ((8*a^6 - 38*a^4*b^2 + 87*a^2*b^4 + 48*b^6)*cos(c + d*x))/(336*b^5*(a^2 - b^2)^4*d*(a + b*sin(c + d*x))) + (5*cos(c + d*x)^3*(2*a + 3*b*sin(c + d*x)))/(42*b^3*d*(a + b*sin(c + d*x))^6) - (cos(c + d*x)*(4*a^2 + 9*b^2 + 10*a*b*sin(c + d*x)))/(42*b^5*d*(a + b*sin(c + d*x))^5)]
    @test_int [cos(c + d*x)^4/(a + b*sin(c + d*x))^8, x, 11, (3*a*(2*a^2 + b^2)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(8*(a^2 - b^2)^(11/2)*d) - cos(c + d*x)^3/(7*b*d*(a + b*sin(c + d*x))^7) - ((a^2 - 3*b^2)*cos(c + d*x))/(140*b^3*(a^2 - b^2)*d*(a + b*sin(c + d*x))^5) - (a*(2*a^2 - 11*b^2)*cos(c + d*x))/(280*b^3*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^4) - ((2*a^4 - 15*a^2*b^2 - 8*b^4)*cos(c + d*x))/(280*b^3*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^3) - (a*(4*a^4 - 36*a^2*b^2 - 73*b^4)*cos(c + d*x))/(560*b^3*(a^2 - b^2)^4*d*(a + b*sin(c + d*x))^2) - ((4*a^6 - 40*a^4*b^2 - 247*a^2*b^4 - 32*b^6)*cos(c + d*x))/(560*b^3*(a^2 - b^2)^5*d*(a + b*sin(c + d*x))) + (cos(c + d*x)*(a + 3*b*sin(c + d*x)))/(28*b^3*d*(a + b*sin(c + d*x))^6)]
    @test_int [cos(c + d*x)^2/(a + b*sin(c + d*x))^8, x, 11, (a*(8*a^4 + 20*a^2*b^2 + 5*b^4)*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(8*(a^2 - b^2)^(13/2)*d) - cos(c + d*x)/(7*b*d*(a + b*sin(c + d*x))^7) + (a*cos(c + d*x))/(42*b*(a^2 - b^2)*d*(a + b*sin(c + d*x))^6) + ((5*a^2 + 6*b^2)*cos(c + d*x))/(210*b*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^5) + (a*(20*a^2 + 79*b^2)*cos(c + d*x))/(840*b*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^4) + ((20*a^4 + 179*a^2*b^2 + 32*b^4)*cos(c + d*x))/(840*b*(a^2 - b^2)^4*d*(a + b*sin(c + d*x))^3) + (a*(40*a^4 + 718*a^2*b^2 + 397*b^4)*cos(c + d*x))/(1680*b*(a^2 - b^2)^5*d*(a + b*sin(c + d*x))^2) + ((40*a^6 + 1518*a^4*b^2 + 1779*a^2*b^4 + 128*b^6)*cos(c + d*x))/(1680*b*(a^2 - b^2)^6*d*(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sin(c + d*x))^8, x, 12, -((9*a*b^2*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(8*(a^2 - b^2)^(17/2)*d)) + (b*sec(c + d*x))/(7*(a^2 - b^2)*d*(a + b*sin(c + d*x))^7) + (5*a*b*sec(c + d*x))/(14*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^6) + (b*(49*a^2 + 16*b^2)*sec(c + d*x))/(70*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^5) + (13*a*b*(28*a^2 + 27*b^2)*sec(c + d*x))/(280*(a^2 - b^2)^4*d*(a + b*sin(c + d*x))^4) + (b*(700*a^4 + 1317*a^2*b^2 + 128*b^4)*sec(c + d*x))/(280*(a^2 - b^2)^5*d*(a + b*sin(c + d*x))^3) + (11*a*b*(280*a^4 + 844*a^2*b^2 + 241*b^4)*sec(c + d*x))/(560*(a^2 - b^2)^6*d*(a + b*sin(c + d*x))^2) + (b*(9800*a^6 + 41484*a^4*b^2 + 22767*a^2*b^4 + 1024*b^6)*sec(c + d*x))/(560*(a^2 - b^2)^7*d*(a + b*sin(c + d*x))) - (sec(c + d*x)*(315*a*b*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6) - (560*a^8 + 42472*a^6*b^2 + 125634*a^4*b^4 + 54511*a^2*b^6 + 2048*b^8)*sin(c + d*x)))/(560*(a^2 - b^2)^8*d)]
    @test_int [sec(c + d*x)^4/(a + b*sin(c + d*x))^8, x, 13, (165*a*b^4*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(8*(a^2 - b^2)^(19/2)*d) + (b*sec(c + d*x)^3)/(7*(a^2 - b^2)*d*(a + b*sin(c + d*x))^7) + (17*a*b*sec(c + d*x)^3)/(42*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^6) + (b*(13*a^2 + 4*b^2)*sec(c + d*x)^3)/(14*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^5) + (a*b*(118*a^2 + 103*b^2)*sec(c + d*x)^3)/(56*(a^2 - b^2)^4*d*(a + b*sin(c + d*x))^4) + (b*(882*a^4 + 1421*a^2*b^2 + 128*b^4)*sec(c + d*x)^3)/(168*(a^2 - b^2)^5*d*(a + b*sin(c + d*x))^3) + (13*a*b*(140*a^4 + 336*a^2*b^2 + 85*b^4)*sec(c + d*x)^3)/(112*(a^2 - b^2)^6*d*(a + b*sin(c + d*x))^2) + (b*(9212*a^6 + 28420*a^4*b^2 + 12907*a^2*b^4 + 512*b^6)*sec(c + d*x)^3)/(112*(a^2 - b^2)^7*d*(a + b*sin(c + d*x))) - (sec(c + d*x)^3*(1155*a*b*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6) - (112*a^8 + 52528*a^6*b^2 + 142902*a^4*b^4 + 57665*a^2*b^6 + 2048*b^8)*sin(c + d*x)))/(336*(a^2 - b^2)^8*d) + (sec(c + d*x)*(3465*a*b^3*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6) + (224*a^10 - 6048*a^8*b^2 - 207332*a^6*b^4 - 413024*a^4*b^6 - 135489*a^2*b^8 - 4096*b^10)*sin(c + d*x)))/(336*(a^2 - b^2)^9*d)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^p*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [cos(c + d*x)^5*sqrt(a + b*sin(c + d*x)), x, 3, (2*(a^2 - b^2)^2*(a + b*sin(c + d*x))^(3/2))/(3*b^5*d) - (8*a*(a^2 - b^2)*(a + b*sin(c + d*x))^(5/2))/(5*b^5*d) + (4*(3*a^2 - b^2)*(a + b*sin(c + d*x))^(7/2))/(7*b^5*d) - (8*a*(a + b*sin(c + d*x))^(9/2))/(9*b^5*d) + (2*(a + b*sin(c + d*x))^(11/2))/(11*b^5*d)]
    @test_int [cos(c + d*x)^3*sqrt(a + b*sin(c + d*x)), x, 3, (-2*(a^2 - b^2)*(a + b*sin(c + d*x))^(3/2))/(3*b^3*d) + (4*a*(a + b*sin(c + d*x))^(5/2))/(5*b^3*d) - (2*(a + b*sin(c + d*x))^(7/2))/(7*b^3*d)]
    @test_int [cos(c + d*x)^1*sqrt(a + b*sin(c + d*x)), x, 2, (2*(a + b*sin(c + d*x))^(3/2))/(3*b*d)]
    @test_int [sec(c + d*x)^1*sqrt(a + b*sin(c + d*x)), x, 5, -((sqrt(a - b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/d) + (sqrt(a + b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/d]
    @test_int [sec(c + d*x)^3*sqrt(a + b*sin(c + d*x)), x, 6, -((2*a - b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(4*sqrt(a - b)*d) + ((2*a + b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(4*sqrt(a + b)*d) + (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^5*sqrt(a + b*sin(c + d*x)), x, 7, -((12*a^2 - 18*a*b + 5*b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(32*(a - b)^(3/2)*d) + ((12*a^2 + 18*a*b + 5*b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(32*(a + b)^(3/2)*d) - (sec(c + d*x)^2*sqrt(a + b*sin(c + d*x))*(a*b - (6*a^2 - 5*b^2)*sin(c + d*x)))/(16*(a^2 - b^2)*d) + (sec(c + d*x)^3*sqrt(a + b*sin(c + d*x))*tan(c + d*x))/(4*d)]

    @test_int [cos(c + d*x)^4*sqrt(a + b*sin(c + d*x)), x, 8, -((4*a*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(21*b*d)) + (2*cos(c + d*x)^3*(a + b*sin(c + d*x))^(3/2))/(9*b*d) - (8*(4*a^4 - 15*a^2*b^2 - 21*b^4)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(315*b^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (32*a*(a^4 - 4*a^2*b^2 + 3*b^4)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(315*b^4*d*sqrt(a + b*sin(c + d*x))) - (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(4*a*(a^2 - 3*b^2) - 3*b*(a^2 + 7*b^2)*sin(c + d*x)))/(315*b^3*d)]
    @test_int [cos(c + d*x)^2*sqrt(a + b*sin(c + d*x)), x, 7, -((4*a*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(15*b*d)) + (2*cos(c + d*x)*(a + b*sin(c + d*x))^(3/2))/(5*b*d) + (4*(a^2 + 3*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (4*a*(a^2 - b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15*b^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2*sqrt(a + b*sin(c + d*x)), x, 7, -((Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(d*sqrt((a + b*sin(c + d*x))/(a + b)))) + (a*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x))) + (sqrt(a + b*sin(c + d*x))*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^4*sqrt(a + b*sin(c + d*x)), x, 7, -(((4*a^2 - 3*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(6*(a^2 - b^2)*d*sqrt((a + b*sin(c + d*x))/(a + b)))) + (2*a*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(a*b - (4*a^2 - 3*b^2)*sin(c + d*x)))/(6*(a^2 - b^2)*d) + (sec(c + d*x)^2*sqrt(a + b*sin(c + d*x))*tan(c + d*x))/(3*d)]

    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x))^(3/2), x, 3, (2*(a^2 - b^2)^2*(a + b*sin(c + d*x))^(5/2))/(5*b^5*d) - (8*a*(a^2 - b^2)*(a + b*sin(c + d*x))^(7/2))/(7*b^5*d) + (4*(3*a^2 - b^2)*(a + b*sin(c + d*x))^(9/2))/(9*b^5*d) - (8*a*(a + b*sin(c + d*x))^(11/2))/(11*b^5*d) + (2*(a + b*sin(c + d*x))^(13/2))/(13*b^5*d)]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x))^(3/2), x, 3, (-2*(a^2 - b^2)*(a + b*sin(c + d*x))^(5/2))/(5*b^3*d) + (4*a*(a + b*sin(c + d*x))^(7/2))/(7*b^3*d) - (2*(a + b*sin(c + d*x))^(9/2))/(9*b^3*d)]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x))^(3/2), x, 2, (2*(a + b*sin(c + d*x))^(5/2))/(5*b*d)]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x))^(3/2), x, 6, -(((a - b)^(3/2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/d) + ((a + b)^(3/2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/d - (2*b*sqrt(a + b*sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x))^(3/2), x, 6, -(sqrt(a - b)*(2*a + b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(4*d) + ((2*a - b)*sqrt(a + b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(4*d) + (sec(c + d*x)^2*(b + a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(2*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x))^(3/2), x, 7, (-3*(4*a^2 - 2*a*b - b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(32*sqrt(a - b)*d) + (3*(4*a^2 + 2*a*b - b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(32*sqrt(a + b)*d) - (sec(c + d*x)^2*(b - 6*a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(16*d) + (sec(c + d*x)^4*(b + a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(4*d)]

    @test_int [cos(c + d*x)^4*(a + b*sin(c + d*x))^(3/2), x, 8, -((2*b*cos(c + d*x)^5*sqrt(a + b*sin(c + d*x)))/(11*d)) - (32*a*(a^4 - 6*a^2*b^2 - 27*b^4)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(1155*b^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (8*(4*a^6 - 25*a^4*b^2 + 6*a^2*b^4 + 15*b^6)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(1155*b^4*d*sqrt(a + b*sin(c + d*x))) + (2*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(a^2 + 3*b^2 + 28*a*b*sin(c + d*x)))/(231*b*d) - (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(4*a^4 - 21*a^2*b^2 - 15*b^4 - 3*a*b*(a^2 + 31*b^2)*sin(c + d*x)))/(1155*b^3*d)]
    @test_int [cos(c + d*x)^2*(a + b*sin(c + d*x))^(3/2), x, 7, -((2*b*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(7*d)) + (4*a*(3*a^2 + 29*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(105*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (4*(3*a^4 + 2*a^2*b^2 - 5*b^4)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(105*b^2*d*sqrt(a + b*sin(c + d*x))) + (2*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(3*a^2 + 5*b^2 + 24*a*b*sin(c + d*x)))/(105*b*d)]
    @test_int [sec(c + d*x)^2*(a + b*sin(c + d*x))^(3/2), x, 6, (sec(c + d*x)*(b + a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/d - (a*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((a^2 - b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*(a + b*sin(c + d*x))^(3/2), x, 7, -((sec(c + d*x)*(b - 4*a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(6*d)) + (sec(c + d*x)^3*(b + a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(3*d) - (2*a*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((4*a^2 - b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(6*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^6*(a + b*sin(c + d*x))^(3/2), x, 8, -((sec(c + d*x)^3*(b - 8*a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(30*d)) + (sec(c + d*x)^5*(b + a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(5*d) - (a*(32*a^2 - 29*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(60*(a^2 - b^2)*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((32*a^2 - 5*b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(60*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(b*(8*a^4 - 13*a^2*b^2 + 5*b^4) - a*(32*a^4 - 61*a^2*b^2 + 29*b^4)*sin(c + d*x)))/(60*(a^2 - b^2)^2*d)]

    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x))^(5/2), x, 3, (2*(a^2 - b^2)^2*(a + b*sin(c + d*x))^(7/2))/(7*b^5*d) - (8*a*(a^2 - b^2)*(a + b*sin(c + d*x))^(9/2))/(9*b^5*d) + (4*(3*a^2 - b^2)*(a + b*sin(c + d*x))^(11/2))/(11*b^5*d) - (8*a*(a + b*sin(c + d*x))^(13/2))/(13*b^5*d) + (2*(a + b*sin(c + d*x))^(15/2))/(15*b^5*d)]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x))^(5/2), x, 3, (-2*(a^2 - b^2)*(a + b*sin(c + d*x))^(7/2))/(7*b^3*d) + (4*a*(a + b*sin(c + d*x))^(9/2))/(9*b^3*d) - (2*(a + b*sin(c + d*x))^(11/2))/(11*b^3*d)]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x))^(5/2), x, 2, (2*(a + b*sin(c + d*x))^(7/2))/(7*b*d)]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x))^(5/2), x, 7, -(((a - b)^(5/2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/d) + ((a + b)^(5/2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/d - (4*a*b*sqrt(a + b*sin(c + d*x)))/d - (2*b*(a + b*sin(c + d*x))^(3/2))/(3*d)]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x))^(5/2), x, 7, -((a - b)^(3/2)*(2*a + 3*b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(4*d) + ((2*a - 3*b)*(a + b)^(3/2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(4*d) + (a*b*sqrt(a + b*sin(c + d*x)))/(2*d) + (sec(c + d*x)^2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^(3/2))/(2*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x))^(5/2), x, 7, (-3*sqrt(a - b)*(4*a^2 + 2*a*b - b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(32*d) + (3*sqrt(a + b)*(4*a^2 - 2*a*b - b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(32*d) + (sec(c + d*x)^4*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^(3/2))/(4*d) + (3*sec(c + d*x)^2*sqrt(a + b*sin(c + d*x))*(a*b + (2*a^2 - b^2)*sin(c + d*x)))/(16*d)]

    @test_int [cos(c + d*x)^4*(a + b*sin(c + d*x))^(5/2), x, 9, -((32*a*b*cos(c + d*x)^5*sqrt(a + b*sin(c + d*x)))/(143*d)) - (2*b*cos(c + d*x)^5*(a + b*sin(c + d*x))^(3/2))/(13*d) - (8*(20*a^6 - 175*a^4*b^2 - 1662*a^2*b^4 - 231*b^6)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15015*b^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (32*a*(5*a^6 - 45*a^4*b^2 - 53*a^2*b^4 + 93*b^6)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15015*b^4*d*sqrt(a + b*sin(c + d*x))) + (2*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(a*(5*a^2 + 59*b^2) + 7*b*(53*a^2 + 11*b^2)*sin(c + d*x)))/(3003*b*d) - (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(4*a*(5*a^4 - 40*a^2*b^2 - 93*b^4) - 3*b*(5*a^4 + 430*a^2*b^2 + 77*b^4)*sin(c + d*x)))/(15015*b^3*d)]
    @test_int [cos(c + d*x)^2*(a + b*sin(c + d*x))^(5/2), x, 8, -((8*a*b*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(21*d)) - (2*b*cos(c + d*x)^3*(a + b*sin(c + d*x))^(3/2))/(9*d) + (4*(5*a^4 + 102*a^2*b^2 + 21*b^4)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(315*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (4*a*(5*a^4 + 22*a^2*b^2 - 27*b^4)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(315*b^2*d*sqrt(a + b*sin(c + d*x))) + (2*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(a*(5*a^2 + 27*b^2) + 3*b*(25*a^2 + 7*b^2)*sin(c + d*x)))/(315*b*d)]
    @test_int [sec(c + d*x)^2*(a + b*sin(c + d*x))^(5/2), x, 7, (a*b*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/d + (sec(c + d*x)*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^(3/2))/d - ((a^2 + 3*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(d*sqrt((a + b*sin(c + d*x))/(a + b))) + (a*(a^2 - b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4*(a + b*sin(c + d*x))^(5/2), x, 7, (sec(c + d*x)^3*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^(3/2))/(3*d) - ((4*a^2 - 3*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(6*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (2*a*(a^2 - b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*d*sqrt(a + b*sin(c + d*x))) + (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(a*b + (4*a^2 - 3*b^2)*sin(c + d*x)))/(6*d)]
    @test_int [sec(c + d*x)^6*(a + b*sin(c + d*x))^(5/2), x, 8, (sec(c + d*x)^5*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^(3/2))/(5*d) - ((32*a^2 - 9*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(60*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (a*(32*a^2 - 17*b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(60*d*sqrt(a + b*sin(c + d*x))) + (sec(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(5*a*b + (8*a^2 - 3*b^2)*sin(c + d*x)))/(30*d) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(8*a*b*(a^2 - b^2) - (32*a^4 - 41*a^2*b^2 + 9*b^4)*sin(c + d*x)))/(60*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^8*(a + b*sin(c + d*x))^(5/2), x, 9, (sec(c + d*x)^7*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^(3/2))/(7*d) - ((128*a^4 - 144*a^2*b^2 + 21*b^4)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(280*(a^2 - b^2)*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (2*a*(8*a^2 - 3*b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(35*d*sqrt(a + b*sin(c + d*x))) + (3*sec(c + d*x)^5*sqrt(a + b*sin(c + d*x))*(3*a*b + (4*a^2 - b^2)*sin(c + d*x)))/(70*d) - (sec(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(4*a*b*(a^2 - b^2) - (32*a^4 - 39*a^2*b^2 + 7*b^4)*sin(c + d*x)))/(140*(a^2 - b^2)*d) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(a*b*(32*a^4 - 59*a^2*b^2 + 27*b^4) - (128*a^6 - 272*a^4*b^2 + 165*a^2*b^4 - 21*b^6)*sin(c + d*x)))/(280*(a^2 - b^2)^2*d)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [cos(c + d*x)^5/sqrt(a + b*sin(c + d*x)), x, 3, (2*(a^2 - b^2)^2*sqrt(a + b*sin(c + d*x)))/(b^5*d) - (8*a*(a^2 - b^2)*(a + b*sin(c + d*x))^(3/2))/(3*b^5*d) + (4*(3*a^2 - b^2)*(a + b*sin(c + d*x))^(5/2))/(5*b^5*d) - (8*a*(a + b*sin(c + d*x))^(7/2))/(7*b^5*d) + (2*(a + b*sin(c + d*x))^(9/2))/(9*b^5*d)]
    @test_int [cos(c + d*x)^3/sqrt(a + b*sin(c + d*x)), x, 3, (-2*(a^2 - b^2)*sqrt(a + b*sin(c + d*x)))/(b^3*d) + (4*a*(a + b*sin(c + d*x))^(3/2))/(3*b^3*d) - (2*(a + b*sin(c + d*x))^(5/2))/(5*b^3*d)]
    @test_int [cos(c + d*x)^1/sqrt(a + b*sin(c + d*x)), x, 2, (2*sqrt(a + b*sin(c + d*x)))/(b*d)]
    @test_int [sec(c + d*x)^1/sqrt(a + b*sin(c + d*x)), x, 5, -(atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b))/(sqrt(a - b)*d)) + atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b))/(sqrt(a + b)*d)]
    @test_int [sec(c + d*x)^3/sqrt(a + b*sin(c + d*x)), x, 6, -((2*a - 3*b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(4*(a - b)^(3/2)*d) + ((2*a + 3*b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(4*(a + b)^(3/2)*d) - (sec(c + d*x)^2*(b - a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^5/sqrt(a + b*sin(c + d*x)), x, 7, (-3*(4*a^2 - 10*a*b + 7*b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(32*(a - b)^(5/2)*d) + (3*(4*a^2 + 10*a*b + 7*b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(32*(a + b)^(5/2)*d) - (sec(c + d*x)^4*(b - a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(4*(a^2 - b^2)*d) - (sec(c + d*x)^2*sqrt(a + b*sin(c + d*x))*(b*(a^2 - 7*b^2) - 6*a*(a^2 - 2*b^2)*sin(c + d*x)))/(16*(a^2 - b^2)^2*d)]

    @test_int [cos(c + d*x)^4/sqrt(a + b*sin(c + d*x)), x, 7, (2*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x)))/(7*b*d) - (32*a*(a^2 - 2*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(35*b^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (8*(4*a^4 - 9*a^2*b^2 + 5*b^4)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(35*b^4*d*sqrt(a + b*sin(c + d*x))) - (4*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(4*a^2 - 5*b^2 - 3*a*b*sin(c + d*x)))/(35*b^3*d)]
    @test_int [cos(c + d*x)^2/sqrt(a + b*sin(c + d*x)), x, 6, (2*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(3*b*d) + (4*a*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (4*(a^2 - b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*b^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/sqrt(a + b*sin(c + d*x)), x, 6, -((sec(c + d*x)*(b - a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/((a^2 - b^2)*d)) - (a*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/((a^2 - b^2)*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^4/sqrt(a + b*sin(c + d*x)), x, 7, -((sec(c + d*x)^3*(b - a*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(3*(a^2 - b^2)*d)) - (2*a*(a^2 - 2*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*(a^2 - b^2)^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((4*a^2 - 5*b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(6*(a^2 - b^2)*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(b*(a^2 - 5*b^2) - 4*a*(a^2 - 2*b^2)*sin(c + d*x)))/(6*(a^2 - b^2)^2*d)]

    @test_int [cos(c + d*x)^5/(a + b*sin(c + d*x))^(3/2), x, 3, (-2*(a^2 - b^2)^2)/(b^5*d*sqrt(a + b*sin(c + d*x))) - (8*a*(a^2 - b^2)*sqrt(a + b*sin(c + d*x)))/(b^5*d) + (4*(3*a^2 - b^2)*(a + b*sin(c + d*x))^(3/2))/(3*b^5*d) - (8*a*(a + b*sin(c + d*x))^(5/2))/(5*b^5*d) + (2*(a + b*sin(c + d*x))^(7/2))/(7*b^5*d)]
    @test_int [cos(c + d*x)^3/(a + b*sin(c + d*x))^(3/2), x, 3, (2*(a^2 - b^2))/(b^3*d*sqrt(a + b*sin(c + d*x))) + (4*a*sqrt(a + b*sin(c + d*x)))/(b^3*d) - (2*(a + b*sin(c + d*x))^(3/2))/(3*b^3*d)]
    @test_int [cos(c + d*x)^1/(a + b*sin(c + d*x))^(3/2), x, 2, -2/(b*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*sin(c + d*x))^(3/2), x, 6, -(atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b))/((a - b)^(3/2)*d)) + atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b))/((a + b)^(3/2)*d) + (2*b)/((a^2 - b^2)*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*sin(c + d*x))^(3/2), x, 7, -((2*a - 5*b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(4*(a - b)^(5/2)*d) + ((2*a + 5*b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(4*(a + b)^(5/2)*d) - (b*(a^2 + 5*b^2))/(2*(a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)^2*(b - a*sin(c + d*x)))/(2*(a^2 - b^2)*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5/(a + b*sin(c + d*x))^(3/2), x, 8, (-3*(4*a^2 - 14*a*b + 15*b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(32*(a - b)^(7/2)*d) + (3*(4*a^2 + 14*a*b + 15*b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(32*(a + b)^(7/2)*d) - (3*b*(2*a^4 - 7*a^2*b^2 - 15*b^4))/(16*(a^2 - b^2)^3*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d*sqrt(a + b*sin(c + d*x))) + (sec(c + d*x)^2*(b*(a^2 + 9*b^2) + 2*a*(3*a^2 - 8*b^2)*sin(c + d*x)))/(16*(a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x)))]

    @test_int [cos(c + d*x)^6/(a + b*sin(c + d*x))^(3/2), x, 8, -((2*cos(c + d*x)^5)/(b*d*sqrt(a + b*sin(c + d*x)))) + (20*cos(c + d*x)^3*(8*a - 7*b*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(63*b^3*d) - (16*(32*a^4 - 57*a^2*b^2 + 21*b^4)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(63*b^6*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (16*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(63*b^6*d*sqrt(a + b*sin(c + d*x))) - (8*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(a*(32*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*sin(c + d*x)))/(63*b^5*d)]
    @test_int [cos(c + d*x)^4/(a + b*sin(c + d*x))^(3/2), x, 7, -((2*cos(c + d*x)^3)/(b*d*sqrt(a + b*sin(c + d*x)))) + (4*cos(c + d*x)*(4*a - 3*b*sin(c + d*x))*sqrt(a + b*sin(c + d*x)))/(5*b^3*d) + (8*(4*a^2 - 3*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(5*b^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (32*a*(a^2 - b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(5*b^4*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sin(c + d*x))^(3/2), x, 6, -((2*cos(c + d*x))/(b*d*sqrt(a + b*sin(c + d*x)))) - (4*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(b^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (4*a*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(b^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sin(c + d*x))^(3/2), x, 7, (2*b*sec(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*sin(c + d*x))) - ((a^2 + 3*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/((a^2 - b^2)^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (a*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/((a^2 - b^2)*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(4*a*b - (a^2 + 3*b^2)*sin(c + d*x)))/((a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^4/(a + b*sin(c + d*x))^(3/2), x, 8, (2*b*sec(c + d*x)^3)/((a^2 - b^2)*d*sqrt(a + b*sin(c + d*x))) - ((4*a^4 - 15*a^2*b^2 - 21*b^4)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(6*(a^2 - b^2)^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (2*a*(a^2 - 3*b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*(a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(8*a*b - (a^2 + 7*b^2)*sin(c + d*x)))/(3*(a^2 - b^2)^2*d) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(a*b*(a^2 - 33*b^2) - (4*a^4 - 15*a^2*b^2 - 21*b^4)*sin(c + d*x)))/(6*(a^2 - b^2)^3*d)]

    @test_int [cos(c + d*x)^5/(a + b*sin(c + d*x))^(5/2), x, 3, (-2*(a^2 - b^2)^2)/(3*b^5*d*(a + b*sin(c + d*x))^(3/2)) + (8*a*(a^2 - b^2))/(b^5*d*sqrt(a + b*sin(c + d*x))) + (4*(3*a^2 - b^2)*sqrt(a + b*sin(c + d*x)))/(b^5*d) - (8*a*(a + b*sin(c + d*x))^(3/2))/(3*b^5*d) + (2*(a + b*sin(c + d*x))^(5/2))/(5*b^5*d)]
    @test_int [cos(c + d*x)^3/(a + b*sin(c + d*x))^(5/2), x, 3, (2*(a^2 - b^2))/(3*b^3*d*(a + b*sin(c + d*x))^(3/2)) - (4*a)/(b^3*d*sqrt(a + b*sin(c + d*x))) - (2*sqrt(a + b*sin(c + d*x)))/(b^3*d)]
    @test_int [cos(c + d*x)^1/(a + b*sin(c + d*x))^(5/2), x, 2, -2/(3*b*d*(a + b*sin(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^1/(a + b*sin(c + d*x))^(5/2), x, 7, -(atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b))/((a - b)^(5/2)*d)) + atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b))/((a + b)^(5/2)*d) + (2*b)/(3*(a^2 - b^2)*d*(a + b*sin(c + d*x))^(3/2)) + (4*a*b)/((a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*sin(c + d*x))^(5/2), x, 8, -((2*a - 7*b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(4*(a - b)^(7/2)*d) + ((2*a + 7*b)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(4*(a + b)^(7/2)*d) - (b*(3*a^2 + 7*b^2))/(6*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^(3/2)) - (sec(c + d*x)^2*(b - a*sin(c + d*x)))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^(3/2)) - (a*b*(a^2 + 19*b^2))/(2*(a^2 - b^2)^3*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^5/(a + b*sin(c + d*x))^(5/2), x, 9, -((12*a^2 - 54*a*b + 77*b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a - b)))/(32*(a - b)^(9/2)*d) + ((12*a^2 + 54*a*b + 77*b^2)*atanh(sqrt(a + b*sin(c + d*x))/sqrt(a + b)))/(32*(a + b)^(9/2)*d) - (b*(18*a^4 - 81*a^2*b^2 - 77*b^4))/(48*(a^2 - b^2)^3*d*(a + b*sin(c + d*x))^(3/2)) - (sec(c + d*x)^4*(b - a*sin(c + d*x)))/(4*(a^2 - b^2)*d*(a + b*sin(c + d*x))^(3/2)) - (a*b*(3*a^4 - 16*a^2*b^2 - 127*b^4))/(8*(a^2 - b^2)^4*d*sqrt(a + b*sin(c + d*x))) + (sec(c + d*x)^2*(b*(3*a^2 + 11*b^2) + 2*a*(3*a^2 - 10*b^2)*sin(c + d*x)))/(16*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^(3/2))]

    @test_int [cos(c + d*x)^8/(a + b*sin(c + d*x))^(5/2), x, 9, -((2*cos(c + d*x)^7)/(3*b*d*(a + b*sin(c + d*x))^(3/2))) - (128*a*(8*a^2 - 9*b^2)*(4*a^2 - 3*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(99*b^8*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (32*(128*a^6 - 272*a^4*b^2 + 159*a^2*b^4 - 15*b^6)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(99*b^8*d*sqrt(a + b*sin(c + d*x))) - (28*cos(c + d*x)^5*(12*a + b*sin(c + d*x)))/(33*b^3*d*sqrt(a + b*sin(c + d*x))) + (40*cos(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(32*a^2 - 3*b^2 - 28*a*b*sin(c + d*x)))/(99*b^5*d) - (16*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(128*a^4 - 144*a^2*b^2 + 15*b^4 - 3*a*b*(32*a^2 - 31*b^2)*sin(c + d*x)))/(99*b^7*d)]
    @test_int [cos(c + d*x)^6/(a + b*sin(c + d*x))^(5/2), x, 8, -((2*cos(c + d*x)^5)/(3*b*d*(a + b*sin(c + d*x))^(3/2))) + (16*a*(32*a^2 - 29*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(21*b^6*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (16*(32*a^4 - 37*a^2*b^2 + 5*b^4)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(21*b^6*d*sqrt(a + b*sin(c + d*x))) - (20*cos(c + d*x)^3*(8*a + b*sin(c + d*x)))/(21*b^3*d*sqrt(a + b*sin(c + d*x))) + (8*cos(c + d*x)*sqrt(a + b*sin(c + d*x))*(32*a^2 - 5*b^2 - 24*a*b*sin(c + d*x)))/(21*b^5*d)]
    @test_int [cos(c + d*x)^4/(a + b*sin(c + d*x))^(5/2), x, 7, -((2*cos(c + d*x)^3)/(3*b*d*(a + b*sin(c + d*x))^(3/2))) - (32*a*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*b^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + (8*(4*a^2 - b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*b^4*d*sqrt(a + b*sin(c + d*x))) - (4*cos(c + d*x)*(4*a + b*sin(c + d*x)))/(3*b^3*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*sin(c + d*x))^(5/2), x, 7, -((2*cos(c + d*x))/(3*b*d*(a + b*sin(c + d*x))^(3/2))) + (4*a*cos(c + d*x))/(3*b*(a^2 - b^2)*d*sqrt(a + b*sin(c + d*x))) + (4*a*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*b^2*(a^2 - b^2)*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (4*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*b^2*d*sqrt(a + b*sin(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*sin(c + d*x))^(5/2), x, 8, (2*b*sec(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sin(c + d*x))^(3/2)) + (16*a*b*sec(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x))) - (a*(3*a^2 + 29*b^2)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*(a^2 - b^2)^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((3*a^2 + 5*b^2)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*(a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(b*(27*a^2 + 5*b^2) - a*(3*a^2 + 29*b^2)*sin(c + d*x)))/(3*(a^2 - b^2)^3*d)]
    @test_int [sec(c + d*x)^4/(a + b*sin(c + d*x))^(5/2), x, 9, (2*b*sec(c + d*x)^3)/(3*(a^2 - b^2)*d*(a + b*sin(c + d*x))^(3/2)) + (8*a*b*sec(c + d*x)^3)/((a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x))) - (2*a*(a^4 - 6*a^2*b^2 - 27*b^4)*Elliptic.E((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*(a^2 - b^2)^4*d*sqrt((a + b*sin(c + d*x))/(a + b))) + ((4*a^4 - 21*a^2*b^2 - 15*b^4)*Elliptic.F((1/2)*(c - pi/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(6*(a^2 - b^2)^3*d*sqrt(a + b*sin(c + d*x))) - (sec(c + d*x)^3*sqrt(a + b*sin(c + d*x))*(b*(29*a^2 + 3*b^2) - a*(a^2 + 31*b^2)*sin(c + d*x)))/(3*(a^2 - b^2)^3*d) - (sec(c + d*x)*sqrt(a + b*sin(c + d*x))*(b*(a^4 - 114*a^2*b^2 - 15*b^4) - 4*a*(a^4 - 6*a^2*b^2 - 27*b^4)*sin(c + d*x)))/(6*(a^2 - b^2)^4*d)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^(p/2)*(a+b*sin(e+f*x))^m=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x)), x, 5, (-2*b*(e*cos(c + d*x))^(9/2))/(9*d*e) + (10*a*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*d*sqrt(e*cos(c + d*x))) + (10*a*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*e*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x)), x, 4, (-2*b*(e*cos(c + d*x))^(7/2))/(7*d*e) + (6*a*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*sqrt(cos(c + d*x))) + (2*a*e*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x)), x, 4, (-2*b*(e*cos(c + d*x))^(5/2))/(5*d*e) + (2*a*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*sqrt(e*cos(c + d*x))) + (2*a*e*sqrt(e*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x)), x, 3, (-2*b*(e*cos(c + d*x))^(3/2))/(3*d*e) + (2*a*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(d*sqrt(cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))/sqrt(e*cos(c + d*x)), x, 3, (-2*b*sqrt(e*cos(c + d*x)))/(d*e) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(d*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))/(e*cos(c + d*x))^(3/2), x, 4, (2*b)/(d*e*sqrt(e*cos(c + d*x))) - (2*a*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(d*e^2*sqrt(cos(c + d*x))) + (2*a*sin(c + d*x))/(d*e*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))/(e*cos(c + d*x))^(5/2), x, 4, (2*b)/(3*d*e*(e*cos(c + d*x))^(3/2)) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*e^2*sqrt(e*cos(c + d*x))) + (2*a*sin(c + d*x))/(3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + b*sin(c + d*x))/(e*cos(c + d*x))^(7/2), x, 5, (2*b)/(5*d*e*(e*cos(c + d*x))^(5/2)) - (6*a*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*e^4*sqrt(cos(c + d*x))) + (2*a*sin(c + d*x))/(5*d*e*(e*cos(c + d*x))^(5/2)) + (6*a*sin(c + d*x))/(5*d*e^3*sqrt(e*cos(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^2, x, 6, (-26*a*b*(e*cos(c + d*x))^(9/2))/(99*d*e) + (10*(11*a^2 + 2*b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(231*d*sqrt(e*cos(c + d*x))) + (10*(11*a^2 + 2*b^2)*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(11*a^2 + 2*b^2)*e*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(77*d) - (2*b*(e*cos(c + d*x))^(9/2)*(a + b*sin(c + d*x)))/(11*d*e)]
    @test_int [(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^2, x, 5, (-22*a*b*(e*cos(c + d*x))^(7/2))/(63*d*e) + (2*(9*a^2 + 2*b^2)*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(15*d*sqrt(cos(c + d*x))) + (2*(9*a^2 + 2*b^2)*e*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(45*d) - (2*b*(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x)))/(9*d*e)]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^2, x, 5, (-18*a*b*(e*cos(c + d*x))^(5/2))/(35*d*e) + (2*(7*a^2 + 2*b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*d*sqrt(e*cos(c + d*x))) + (2*(7*a^2 + 2*b^2)*e*sqrt(e*cos(c + d*x))*sin(c + d*x))/(21*d) - (2*b*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x)))/(7*d*e)]
    @test_int [sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^2, x, 4, (-14*a*b*(e*cos(c + d*x))^(3/2))/(15*d*e) + (2*(5*a^2 + 2*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*sqrt(cos(c + d*x))) - (2*b*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x)))/(5*d*e)]
    @test_int [(a + b*sin(c + d*x))^2/sqrt(e*cos(c + d*x)), x, 4, (-10*a*b*sqrt(e*cos(c + d*x)))/(3*d*e) + (2*(3*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*sqrt(e*cos(c + d*x))) - (2*b*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x)))/(3*d*e)]
    @test_int [(a + b*sin(c + d*x))^2/(e*cos(c + d*x))^(3/2), x, 4, (2*a*b*(e*cos(c + d*x))^(3/2))/(d*e^3) - (2*(a^2 + 2*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(d*e^2*sqrt(cos(c + d*x))) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(d*e*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))^2/(e*cos(c + d*x))^(5/2), x, 4, (2*a*b*sqrt(e*cos(c + d*x)))/(3*d*e^3) + (2*(a^2 - 2*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*e^2*sqrt(e*cos(c + d*x))) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + b*sin(c + d*x))^2/(e*cos(c + d*x))^(7/2), x, 5, (2*a*b)/(5*d*e^3*sqrt(e*cos(c + d*x))) - (2*(3*a^2 - 2*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*e^4*sqrt(cos(c + d*x))) + (2*(3*a^2 - 2*b^2)*sin(c + d*x))/(5*d*e^3*sqrt(e*cos(c + d*x))) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x)))/(5*d*e*(e*cos(c + d*x))^(5/2))]

    @test_int [(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^3, x, 7, (-2*b*(177*a^2 + 44*b^2)*(e*cos(c + d*x))^(9/2))/(1287*d*e) + (10*a*(11*a^2 + 6*b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(231*d*sqrt(e*cos(c + d*x))) + (10*a*(11*a^2 + 6*b^2)*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(231*d) + (2*a*(11*a^2 + 6*b^2)*e*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(77*d) - (34*a*b*(e*cos(c + d*x))^(9/2)*(a + b*sin(c + d*x)))/(143*d*e) - (2*b*(e*cos(c + d*x))^(9/2)*(a + b*sin(c + d*x))^2)/(13*d*e)]
    @test_int [(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^3, x, 6, (-2*b*(43*a^2 + 12*b^2)*(e*cos(c + d*x))^(7/2))/(231*d*e) + (2*a*(3*a^2 + 2*b^2)*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*sqrt(cos(c + d*x))) + (2*a*(3*a^2 + 2*b^2)*e*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(15*d) - (10*a*b*(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x)))/(33*d*e) - (2*b*(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^2)/(11*d*e)]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^3, x, 6, (-2*b*(89*a^2 + 28*b^2)*(e*cos(c + d*x))^(5/2))/(315*d*e) + (2*a*(7*a^2 + 6*b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*d*sqrt(e*cos(c + d*x))) + (2*a*(7*a^2 + 6*b^2)*e*sqrt(e*cos(c + d*x))*sin(c + d*x))/(21*d) - (26*a*b*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x)))/(63*d*e) - (2*b*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^2)/(9*d*e)]
    @test_int [sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^3, x, 5, (-2*b*(57*a^2 + 20*b^2)*(e*cos(c + d*x))^(3/2))/(105*d*e) + (2*a*(5*a^2 + 6*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*sqrt(cos(c + d*x))) - (22*a*b*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x)))/(35*d*e) - (2*b*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^2)/(7*d*e)]
    @test_int [(a + b*sin(c + d*x))^3/sqrt(e*cos(c + d*x)), x, 5, (-2*b*(11*a^2 + 4*b^2)*sqrt(e*cos(c + d*x)))/(5*d*e) + (2*a*(a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(d*sqrt(e*cos(c + d*x))) - (6*a*b*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x)))/(5*d*e) - (2*b*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^2)/(5*d*e)]
    @test_int [(a + b*sin(c + d*x))^3/(e*cos(c + d*x))^(3/2), x, 5, (2*b*(3*a^2 + 4*b^2)*(e*cos(c + d*x))^(3/2))/(3*d*e^3) - (2*a*(a^2 + 6*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(d*e^2*sqrt(cos(c + d*x))) + (2*a*b*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x)))/(d*e^3) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(d*e*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))^3/(e*cos(c + d*x))^(5/2), x, 5, (2*b*(a^2 + 4*b^2)*sqrt(e*cos(c + d*x)))/(3*d*e^3) + (2*a*(a^2 - 6*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*e^2*sqrt(e*cos(c + d*x))) + (2*a*b*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x)))/(3*d*e^3) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + b*sin(c + d*x))^3/(e*cos(c + d*x))^(7/2), x, 5, (2*b*(3*a^2 - 4*b^2)*(e*cos(c + d*x))^(3/2))/(5*d*e^5) - (6*a*(a^2 - 2*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*e^4*sqrt(cos(c + d*x))) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(5*d*e*(e*cos(c + d*x))^(5/2)) - (2*(a + b*sin(c + d*x))*(a*b - (3*a^2 - 4*b^2)*sin(c + d*x)))/(5*d*e^3*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))^3/(e*cos(c + d*x))^(9/2), x, 5, (2*b*(5*a^2 - 4*b^2)*sqrt(e*cos(c + d*x)))/(21*d*e^5) + (2*a*(5*a^2 - 6*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*d*e^4*sqrt(e*cos(c + d*x))) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^2)/(7*d*e*(e*cos(c + d*x))^(7/2)) + (2*(a + b*sin(c + d*x))*(a*b + (5*a^2 - 4*b^2)*sin(c + d*x)))/(21*d*e^3*(e*cos(c + d*x))^(3/2))]

    @test_int [(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^4, x, 8, (-34*a*b*(53*a^2 + 38*b^2)*(e*cos(c + d*x))^(9/2))/(6435*d*e) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(231*d*sqrt(e*cos(c + d*x))) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^3*sqrt(e*cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e*(e*cos(c + d*x))^(5/2)*sin(c + d*x))/(385*d) - (2*b*(93*a^2 + 26*b^2)*(e*cos(c + d*x))^(9/2)*(a + b*sin(c + d*x)))/(715*d*e) - (14*a*b*(e*cos(c + d*x))^(9/2)*(a + b*sin(c + d*x))^2)/(65*d*e) - (2*b*(e*cos(c + d*x))^(9/2)*(a + b*sin(c + d*x))^3)/(15*d*e)]
    @test_int [(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^4, x, 7, (-10*a*b*(115*a^2 + 94*b^2)*(e*cos(c + d*x))^(7/2))/(3003*d*e) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(65*d*sqrt(cos(c + d*x))) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e*(e*cos(c + d*x))^(3/2)*sin(c + d*x))/(195*d) - (2*b*(73*a^2 + 22*b^2)*(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x)))/(429*d*e) - (38*a*b*(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^2)/(143*d*e) - (2*b*(e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^3)/(13*d*e)]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^4, x, 7, (-26*a*b*(79*a^2 + 74*b^2)*(e*cos(c + d*x))^(5/2))/(3465*d*e) + (2*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(231*d*sqrt(e*cos(c + d*x))) + (2*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e*sqrt(e*cos(c + d*x))*sin(c + d*x))/(231*d) - (2*b*(167*a^2 + 54*b^2)*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x)))/(693*d*e) - (34*a*b*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^2)/(99*d*e) - (2*b*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^3)/(11*d*e)]
    @test_int [sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^4, x, 6, (-22*a*b*(17*a^2 + 18*b^2)*(e*cos(c + d*x))^(3/2))/(315*d*e) + (2*(15*a^4 + 36*a^2*b^2 + 4*b^4)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(15*d*sqrt(cos(c + d*x))) - (2*b*(41*a^2 + 14*b^2)*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x)))/(105*d*e) - (10*a*b*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^2)/(21*d*e) - (2*b*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^3)/(9*d*e)]
    @test_int [(a + b*sin(c + d*x))^4/sqrt(e*cos(c + d*x)), x, 6, (-6*a*b*(31*a^2 + 34*b^2)*sqrt(e*cos(c + d*x)))/(35*d*e) + (2*(7*a^4 + 28*a^2*b^2 + 4*b^4)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(7*d*sqrt(e*cos(c + d*x))) - (2*b*(29*a^2 + 10*b^2)*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x)))/(35*d*e) - (26*a*b*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^2)/(35*d*e) - (2*b*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^3)/(7*d*e)]
    @test_int [(a + b*sin(c + d*x))^4/(e*cos(c + d*x))^(3/2), x, 6, (2*a*b*(15*a^2 + 62*b^2)*(e*cos(c + d*x))^(3/2))/(15*d*e^3) - (2*(5*a^4 + 60*a^2*b^2 + 12*b^4)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*e^2*sqrt(cos(c + d*x))) + (2*b*(5*a^2 + 6*b^2)*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x)))/(5*d*e^3) + (2*a*b*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^2)/(d*e^3) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^3)/(d*e*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))^4/(e*cos(c + d*x))^(5/2), x, 6, (2*a*b*(a^2 + 14*b^2)*sqrt(e*cos(c + d*x)))/(3*d*e^3) + (2*(a^4 - 12*a^2*b^2 - 4*b^4)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*d*e^2*sqrt(e*cos(c + d*x))) + (2*b*(a^2 + 2*b^2)*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x)))/(3*d*e^3) + (2*a*b*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^2)/(3*d*e^3) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^3)/(3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + b*sin(c + d*x))^4/(e*cos(c + d*x))^(7/2), x, 6, (2*a*b*(3*a^2 - 10*b^2)*(e*cos(c + d*x))^(3/2))/(5*d*e^5) - (6*(a^4 - 4*a^2*b^2 - 4*b^4)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*d*e^4*sqrt(cos(c + d*x))) + (6*b*(a^2 - 2*b^2)*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x)))/(5*d*e^5) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^3)/(5*d*e*(e*cos(c + d*x))^(5/2)) - (6*(a + b*sin(c + d*x))^2*(a*b - (a^2 - 2*b^2)*sin(c + d*x)))/(5*d*e^3*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))^4/(e*cos(c + d*x))^(9/2), x, 6, (10*a*b*(a^2 - 2*b^2)*sqrt(e*cos(c + d*x)))/(21*d*e^5) + (2*(5*a^4 - 12*a^2*b^2 + 12*b^4)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*d*e^4*sqrt(e*cos(c + d*x))) + (2*b*(5*a^2 - 6*b^2)*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x)))/(21*d*e^5) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^3)/(7*d*e*(e*cos(c + d*x))^(7/2)) - (2*(a + b*sin(c + d*x))^2*(a*b - (5*a^2 - 6*b^2)*sin(c + d*x)))/(21*d*e^3*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + b*sin(c + d*x))^4/(e*cos(c + d*x))^(11/2), x, 6, (2*a*b*(21*a^2 - 22*b^2)*(e*cos(c + d*x))^(3/2))/(45*d*e^7) - (2*(7*a^4 - 12*a^2*b^2 + 4*b^4)*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(15*d*e^6*sqrt(cos(c + d*x))) + (2*(b + a*sin(c + d*x))*(a + b*sin(c + d*x))^3)/(9*d*e*(e*cos(c + d*x))^(9/2)) - (2*(a + b*sin(c + d*x))*(b*(7*a^2 - 6*b^2) - a*(21*a^2 - 22*b^2)*sin(c + d*x)))/(45*d*e^5*sqrt(e*cos(c + d*x))) + (2*(a + b*sin(c + d*x))^2*(a*b + (7*a^2 - 6*b^2)*sin(c + d*x)))/(45*d*e^3*(e*cos(c + d*x))^(5/2))]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [(e*cos(c + d*x))^(11/2)/(a + b*sin(c + d*x)), x, 15, -(((-a^2 + b^2)^(9/4)*e^(11/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(11/2)*d)) - ((-a^2 + b^2)^(9/4)*e^(11/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(11/2)*d) + (2*e*(e*cos(c + d*x))^(9/2))/(9*b*d) + (2*a*(21*a^4 - 49*a^2*b^2 + 33*b^4)*e^6*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(21*b^6*d*sqrt(e*cos(c + d*x))) - (a*(a^2 - b^2)^3*e^6*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^6*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (a*(a^2 - b^2)^3*e^6*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^6*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (2*e^3*(e*cos(c + d*x))^(5/2)*(7*(a^2 - b^2) - 5*a*b*sin(c + d*x)))/(35*b^3*d) + (2*e^5*sqrt(e*cos(c + d*x))*(21*(a^2 - b^2)^2 - a*b*(7*a^2 - 12*b^2)*sin(c + d*x)))/(21*b^5*d)]
    @test_int [(e*cos(c + d*x))^(9/2)/(a + b*sin(c + d*x)), x, 14, ((-a^2 + b^2)^(7/4)*e^(9/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(9/2)*d) - ((-a^2 + b^2)^(7/4)*e^(9/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(9/2)*d) + (2*e*(e*cos(c + d*x))^(7/2))/(7*b*d) - (2*a*(5*a^2 - 8*b^2)*e^4*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*b^4*d*sqrt(cos(c + d*x))) + (a*(a^2 - b^2)^2*e^5*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^5*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (a*(a^2 - b^2)^2*e^5*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^5*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (2*e^3*(e*cos(c + d*x))^(3/2)*(5*(a^2 - b^2) - 3*a*b*sin(c + d*x)))/(15*b^3*d)]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + b*sin(c + d*x)), x, 14, -(((-a^2 + b^2)^(5/4)*e^(7/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(7/2)*d)) - ((-a^2 + b^2)^(5/4)*e^(7/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(7/2)*d) + (2*e*(e*cos(c + d*x))^(5/2))/(5*b*d) - (2*a*(3*a^2 - 4*b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*b^4*d*sqrt(e*cos(c + d*x))) + (a*(a^2 - b^2)^2*e^4*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^4*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (a*(a^2 - b^2)^2*e^4*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^4*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (2*e^3*sqrt(e*cos(c + d*x))*(3*(a^2 - b^2) - a*b*sin(c + d*x)))/(3*b^3*d)]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + b*sin(c + d*x)), x, 13, ((-a^2 + b^2)^(3/4)*e^(5/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(5/2)*d) - ((-a^2 + b^2)^(3/4)*e^(5/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(5/2)*d) + (2*e*(e*cos(c + d*x))^(3/2))/(3*b*d) + (2*a*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(b^2*d*sqrt(cos(c + d*x))) - (a*(a^2 - b^2)*e^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^3*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (a*(a^2 - b^2)*e^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^3*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + b*sin(c + d*x)), x, 13, -(((-a^2 + b^2)^(1/4)*e^(3/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(3/2)*d)) - ((-a^2 + b^2)^(1/4)*e^(3/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(b^(3/2)*d) + (2*e*sqrt(e*cos(c + d*x)))/(b*d) + (2*a*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(b^2*d*sqrt(e*cos(c + d*x))) - (a*(a^2 - b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^2*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (a*(a^2 - b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b^2*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + b*sin(c + d*x)), x, 9, (sqrt(e)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(sqrt(b)*(-a^2 + b^2)^(1/4)*d) - (sqrt(e)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(sqrt(b)*(-a^2 + b^2)^(1/4)*d) + (a*e*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (a*e*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(b*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))), x, 9, -((sqrt(b)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/((-a^2 + b^2)^(3/4)*d*sqrt(e))) - (sqrt(b)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/((-a^2 + b^2)^(3/4)*d*sqrt(e)) + (a*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/((a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (a*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/((a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))), x, 13, (b^(3/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/((-a^2 + b^2)^(5/4)*d*e^(3/2)) - (b^(3/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/((-a^2 + b^2)^(5/4)*d*e^(3/2)) - (2*a*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/((a^2 - b^2)*d*e^2*sqrt(cos(c + d*x))) - (a*b*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/((a^2 - b^2)*(b - sqrt(-a^2 + b^2))*d*e*sqrt(e*cos(c + d*x))) - (a*b*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/((a^2 - b^2)*(b + sqrt(-a^2 + b^2))*d*e*sqrt(e*cos(c + d*x))) - (2*(b - a*sin(c + d*x)))/((a^2 - b^2)*d*e*sqrt(e*cos(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))), x, 13, -((b^(5/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/((-a^2 + b^2)^(7/4)*d*e^(5/2))) - (b^(5/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/((-a^2 + b^2)^(7/4)*d*e^(5/2)) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*(a^2 - b^2)*d*e^2*sqrt(e*cos(c + d*x))) - (a*b^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/((a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*e^2*sqrt(e*cos(c + d*x))) - (a*b^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/((a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*e^2*sqrt(e*cos(c + d*x))) - (2*(b - a*sin(c + d*x)))/(3*(a^2 - b^2)*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [1/((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))), x, 14, (b^(7/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/((-a^2 + b^2)^(9/4)*d*e^(7/2)) - (b^(7/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/((-a^2 + b^2)^(9/4)*d*e^(7/2)) - (2*a*(3*a^2 - 8*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*(a^2 - b^2)^2*d*e^4*sqrt(cos(c + d*x))) + (a*b^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/((a^2 - b^2)^2*(b - sqrt(-a^2 + b^2))*d*e^3*sqrt(e*cos(c + d*x))) + (a*b^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/((a^2 - b^2)^2*(b + sqrt(-a^2 + b^2))*d*e^3*sqrt(e*cos(c + d*x))) - (2*(b - a*sin(c + d*x)))/(5*(a^2 - b^2)*d*e*(e*cos(c + d*x))^(5/2)) + (2*(5*b^3 + a*(3*a^2 - 8*b^2)*sin(c + d*x)))/(5*(a^2 - b^2)^2*d*e^3*sqrt(e*cos(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(11/2)/(a + b*sin(c + d*x))^2, x, 15, (-9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(11/2)*d) - (9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(11/2)*d) - (3*(21*a^4 - 28*a^2*b^2 + 5*b^4)*e^6*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(7*b^6*d*sqrt(e*cos(c + d*x))) + (9*a^2*(a^2 - b^2)^2*e^6*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^6*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (9*a^2*(a^2 - b^2)^2*e^6*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^6*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (9*e^3*(e*cos(c + d*x))^(5/2)*(7*a - 5*b*sin(c + d*x)))/(35*b^3*d) - (e*(e*cos(c + d*x))^(9/2))/(b*d*(a + b*sin(c + d*x))) - (3*e^5*sqrt(e*cos(c + d*x))*(21*a*(a^2 - b^2) - b*(7*a^2 - 5*b^2)*sin(c + d*x)))/(7*b^5*d)]
    @test_int [(e*cos(c + d*x))^(9/2)/(a + b*sin(c + d*x))^2, x, 14, (7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(9/2)*d) - (7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(9/2)*d) + (7*(5*a^2 - 3*b^2)*e^4*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*b^4*d*sqrt(cos(c + d*x))) - (7*a^2*(a^2 - b^2)*e^5*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^5*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (7*a^2*(a^2 - b^2)*e^5*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^5*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (7*e^3*(e*cos(c + d*x))^(3/2)*(5*a - 3*b*sin(c + d*x)))/(15*b^3*d) - (e*(e*cos(c + d*x))^(7/2))/(b*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + b*sin(c + d*x))^2, x, 14, (-5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(7/2)*d) - (5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(7/2)*d) + (5*(3*a^2 - b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*b^4*d*sqrt(e*cos(c + d*x))) - (5*a^2*(a^2 - b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^4*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (5*a^2*(a^2 - b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^4*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (5*e^3*sqrt(e*cos(c + d*x))*(3*a - b*sin(c + d*x)))/(3*b^3*d) - (e*(e*cos(c + d*x))^(5/2))/(b*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + b*sin(c + d*x))^2, x, 13, (3*a*e^(5/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) - (3*a*e^(5/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) - (3*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(b^2*d*sqrt(cos(c + d*x))) + (3*a^2*e^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^3*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (3*a^2*e^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^3*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(3/2))/(b*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + b*sin(c + d*x))^2, x, 13, -(a*e^(3/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) - (a*e^(3/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) - (e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(b^2*d*sqrt(e*cos(c + d*x))) + (a^2*e^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^2*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (a^2*e^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b^2*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (e*sqrt(e*cos(c + d*x)))/(b*d*(a + b*sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + b*sin(c + d*x))^2, x, 13, -(a*sqrt(e)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*sqrt(b)*(-a^2 + b^2)^(5/4)*d) + (a*sqrt(e)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*sqrt(b)*(-a^2 + b^2)^(5/4)*d) + (sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/((a^2 - b^2)*d*sqrt(cos(c + d*x))) + (a^2*e*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b*(a^2 - b^2)*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (a^2*e*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*b*(a^2 - b^2)*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (b*(e*cos(c + d*x))^(3/2))/((a^2 - b^2)*d*e*(a + b*sin(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^2), x, 13, (3*a*sqrt(b)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*(-a^2 + b^2)^(7/4)*d*sqrt(e)) + (3*a*sqrt(b)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*(-a^2 + b^2)^(7/4)*d*sqrt(e)) - (sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/((a^2 - b^2)*d*sqrt(e*cos(c + d*x))) + (3*a^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*(a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (3*a^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*(a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (b*sqrt(e*cos(c + d*x)))/((a^2 - b^2)*d*e*(a + b*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^2), x, 14, (-5*a*b^(3/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) + (5*a*b^(3/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) - ((2*a^2 + 3*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/((a^2 - b^2)^2*d*e^2*sqrt(cos(c + d*x))) - (5*a^2*b*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*(a^2 - b^2)^2*(b - sqrt(-a^2 + b^2))*d*e*sqrt(e*cos(c + d*x))) - (5*a^2*b*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*(a^2 - b^2)^2*(b + sqrt(-a^2 + b^2))*d*e*sqrt(e*cos(c + d*x))) + b/((a^2 - b^2)*d*e*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))) - (5*a*b - (2*a^2 + 3*b^2)*sin(c + d*x))/((a^2 - b^2)^2*d*e*sqrt(e*cos(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^2), x, 14, (7*a*b^(5/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) + (7*a*b^(5/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) + ((2*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(3*(a^2 - b^2)^2*d*e^2*sqrt(e*cos(c + d*x))) - (7*a^2*b^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*(a^2 - b^2)^2*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*e^2*sqrt(e*cos(c + d*x))) - (7*a^2*b^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*(a^2 - b^2)^2*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*e^2*sqrt(e*cos(c + d*x))) + b/((a^2 - b^2)*d*e*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))) - (7*a*b - (2*a^2 + 5*b^2)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [1/((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^2), x, 15, (-9*a*b^(7/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) + (9*a*b^(7/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) - (3*(2*a^4 - 10*a^2*b^2 - 7*b^4)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(5*(a^2 - b^2)^3*d*e^4*sqrt(cos(c + d*x))) + (9*a^2*b^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*(a^2 - b^2)^3*(b - sqrt(-a^2 + b^2))*d*e^3*sqrt(e*cos(c + d*x))) + (9*a^2*b^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(2*(a^2 - b^2)^3*(b + sqrt(-a^2 + b^2))*d*e^3*sqrt(e*cos(c + d*x))) + b/((a^2 - b^2)*d*e*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))) - (9*a*b - (2*a^2 + 7*b^2)*sin(c + d*x))/(5*(a^2 - b^2)^2*d*e*(e*cos(c + d*x))^(5/2)) + (3*(15*a*b^3 + (2*a^4 - 10*a^2*b^2 - 7*b^4)*sin(c + d*x)))/(5*(a^2 - b^2)^3*d*e^3*sqrt(e*cos(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(13/2)/(a + b*sin(c + d*x))^3, x, 15, (-11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) + (11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) + (11*a*(45*a^2 - 37*b^2)*e^6*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(20*b^6*d*sqrt(cos(c + d*x))) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^7*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^7*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(11/2))/(2*b*d*(a + b*sin(c + d*x))^2) - (11*e^3*(e*cos(c + d*x))^(7/2)*(9*a + 2*b*sin(c + d*x)))/(28*b^3*d*(a + b*sin(c + d*x))) + (11*e^5*(e*cos(c + d*x))^(3/2)*(5*(9*a^2 - 2*b^2) - 27*a*b*sin(c + d*x)))/(60*b^5*d)]
    @test_int [(e*cos(c + d*x))^(11/2)/(a + b*sin(c + d*x))^3, x, 15, (9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (3*a*(21*a^2 - 13*b^2)*e^6*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(4*b^6*d*sqrt(e*cos(c + d*x))) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^6*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^6*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(9/2))/(2*b*d*(a + b*sin(c + d*x))^2) - (9*e^3*(e*cos(c + d*x))^(5/2)*(7*a + 2*b*sin(c + d*x)))/(20*b^3*d*(a + b*sin(c + d*x))) + (3*e^5*sqrt(e*cos(c + d*x))*(3*(7*a^2 - 2*b^2) - 7*a*b*sin(c + d*x)))/(4*b^5*d)]
    @test_int [(e*cos(c + d*x))^(9/2)/(a + b*sin(c + d*x))^3, x, 14, (7*(5*a^2 - 2*b^2)*e^(9/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) - (7*(5*a^2 - 2*b^2)*e^(9/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) - (35*a*e^4*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(4*b^4*d*sqrt(cos(c + d*x))) + (7*a*(5*a^2 - 2*b^2)*e^5*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^5*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (7*a*(5*a^2 - 2*b^2)*e^5*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^5*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(7/2))/(2*b*d*(a + b*sin(c + d*x))^2) - (7*e^3*(e*cos(c + d*x))^(3/2)*(5*a + 2*b*sin(c + d*x)))/(12*b^3*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + b*sin(c + d*x))^3, x, 14, (-5*(3*a^2 - 2*b^2)*e^(7/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) - (5*(3*a^2 - 2*b^2)*e^(7/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) - (15*a*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(4*b^4*d*sqrt(e*cos(c + d*x))) + (5*a*(3*a^2 - 2*b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^4*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (5*a*(3*a^2 - 2*b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^4*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(5/2))/(2*b*d*(a + b*sin(c + d*x))^2) - (5*e^3*sqrt(e*cos(c + d*x))*(3*a + 2*b*sin(c + d*x)))/(4*b^3*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + b*sin(c + d*x))^3, x, 14, (3*(a^2 - 2*b^2)*e^(5/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) - (3*(a^2 - 2*b^2)*e^(5/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) + (3*a*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(4*b^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))) - (3*a*(a^2 - 2*b^2)*e^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^3*(a^2 - b^2)*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (3*a*(a^2 - 2*b^2)*e^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^3*(a^2 - b^2)*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(3/2))/(2*b*d*(a + b*sin(c + d*x))^2) + (3*a*e*(e*cos(c + d*x))^(3/2))/(4*b*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + b*sin(c + d*x))^3, x, 14, ((a^2 + 2*b^2)*e^(3/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) + ((a^2 + 2*b^2)*e^(3/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) - (a*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(4*b^2*(a^2 - b^2)*d*sqrt(e*cos(c + d*x))) + (a*(a^2 + 2*b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^2*(a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (a*(a^2 + 2*b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b^2*(a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (e*sqrt(e*cos(c + d*x)))/(2*b*d*(a + b*sin(c + d*x))^2) + (a*e*sqrt(e*cos(c + d*x)))/(4*b*(a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + b*sin(c + d*x))^3, x, 14, ((3*a^2 + 2*b^2)*sqrt(e)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*sqrt(b)*(-a^2 + b^2)^(9/4)*d) - ((3*a^2 + 2*b^2)*sqrt(e)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*sqrt(b)*(-a^2 + b^2)^(9/4)*d) + (5*a*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + (a*(3*a^2 + 2*b^2)*e*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b*(a^2 - b^2)^2*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (a*(3*a^2 + 2*b^2)*e*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*b*(a^2 - b^2)^2*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (b*(e*cos(c + d*x))^(3/2))/(2*(a^2 - b^2)*d*e*(a + b*sin(c + d*x))^2) + (5*a*b*(e*cos(c + d*x))^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*sin(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^3), x, 14, (-3*sqrt(b)*(5*a^2 + 2*b^2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*(-a^2 + b^2)^(11/4)*d*sqrt(e)) - (3*sqrt(b)*(5*a^2 + 2*b^2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*(-a^2 + b^2)^(11/4)*d*sqrt(e)) - (7*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(4*(a^2 - b^2)^2*d*sqrt(e*cos(c + d*x))) + (3*a*(5*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*(a^2 - b^2)^2*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (3*a*(5*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*(a^2 - b^2)^2*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (b*sqrt(e*cos(c + d*x)))/(2*(a^2 - b^2)*d*e*(a + b*sin(c + d*x))^2) + (7*a*b*sqrt(e*cos(c + d*x)))/(4*(a^2 - b^2)^2*d*e*(a + b*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^3), x, 15, (5*b^(3/2)*(7*a^2 + 2*b^2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - (5*b^(3/2)*(7*a^2 + 2*b^2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - (a*(8*a^2 + 37*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(4*(a^2 - b^2)^3*d*e^2*sqrt(cos(c + d*x))) - (5*a*b*(7*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*(a^2 - b^2)^3*(b - sqrt(-a^2 + b^2))*d*e*sqrt(e*cos(c + d*x))) - (5*a*b*(7*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*(a^2 - b^2)^3*(b + sqrt(-a^2 + b^2))*d*e*sqrt(e*cos(c + d*x))) + b/(2*(a^2 - b^2)*d*e*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^2) + (9*a*b)/(4*(a^2 - b^2)^2*d*e*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))) - (5*b*(7*a^2 + 2*b^2) - a*(8*a^2 + 37*b^2)*sin(c + d*x))/(4*(a^2 - b^2)^3*d*e*sqrt(e*cos(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^3), x, 15, (-7*b^(5/2)*(9*a^2 + 2*b^2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) - (7*b^(5/2)*(9*a^2 + 2*b^2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) + (a*(8*a^2 + 69*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(12*(a^2 - b^2)^3*d*e^2*sqrt(e*cos(c + d*x))) - (7*a*b^2*(9*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*(a^2 - b^2)^3*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*e^2*sqrt(e*cos(c + d*x))) - (7*a*b^2*(9*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*(a^2 - b^2)^3*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*e^2*sqrt(e*cos(c + d*x))) + b/(2*(a^2 - b^2)*d*e*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^2) + (11*a*b)/(4*(a^2 - b^2)^2*d*e*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))) - (7*b*(9*a^2 + 2*b^2) - a*(8*a^2 + 69*b^2)*sin(c + d*x))/(12*(a^2 - b^2)^3*d*e*(e*cos(c + d*x))^(3/2))]
    @test_int [1/((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^3), x, 16, (9*b^(7/2)*(11*a^2 + 2*b^2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - (9*b^(7/2)*(11*a^2 + 2*b^2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - (3*a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(20*(a^2 - b^2)^4*d*e^4*sqrt(cos(c + d*x))) + (9*a*b^3*(11*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*(a^2 - b^2)^4*(b - sqrt(-a^2 + b^2))*d*e^3*sqrt(e*cos(c + d*x))) + (9*a*b^3*(11*a^2 + 2*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(8*(a^2 - b^2)^4*(b + sqrt(-a^2 + b^2))*d*e^3*sqrt(e*cos(c + d*x))) + b/(2*(a^2 - b^2)*d*e*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^2) + (13*a*b)/(4*(a^2 - b^2)^2*d*e*(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))) - (9*b*(11*a^2 + 2*b^2) - a*(8*a^2 + 109*b^2)*sin(c + d*x))/(20*(a^2 - b^2)^3*d*e*(e*cos(c + d*x))^(5/2)) + (3*(15*b^3*(11*a^2 + 2*b^2) + a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*sin(c + d*x)))/(20*(a^2 - b^2)^4*d*e^3*sqrt(e*cos(c + d*x)))]

    @test_int [(e*cos(c + d*x))^(15/2)/(a + b*sin(c + d*x))^4, x, 16, (39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (13*(231*a^4 - 203*a^2*b^2 + 20*b^4)*e^8*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(56*b^8*d*sqrt(e*cos(c + d*x))) - (39*a^2*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^8*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(c + d*x), 2))/(16*b^8*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (39*a^2*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^8*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(c + d*x), 2))/(16*b^8*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(13/2))/(3*b*d*(a + b*sin(c + d*x))^3) - (13*e^3*(e*cos(c + d*x))^(9/2)*(11*a + 4*b*sin(c + d*x)))/(84*b^3*d*(a + b*sin(c + d*x))^2) - (39*e^5*(e*cos(c + d*x))^(5/2)*(77*a^2 - 20*b^2 + 22*a*b*sin(c + d*x)))/(280*b^5*d*(a + b*sin(c + d*x))) + (13*e^7*sqrt(e*cos(c + d*x))*(21*a*(11*a^2 - 6*b^2) - b*(77*a^2 - 20*b^2)*sin(c + d*x)))/(56*b^7*d)]
    @test_int [(e*cos(c + d*x))^(13/2)/(a + b*sin(c + d*x))^4, x, 15, (77*a*(3*a^2 - 2*b^2)*e^(13/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (77*a*(3*a^2 - 2*b^2)*e^(13/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (77*(15*a^2 - 4*b^2)*e^6*sqrt(e*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(40*b^6*d*sqrt(cos(c + d*x))) + (77*a^2*(3*a^2 - 2*b^2)*e^7*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(c + d*x), 2))/(16*b^7*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (77*a^2*(3*a^2 - 2*b^2)*e^7*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(c + d*x), 2))/(16*b^7*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(11/2))/(3*b*d*(a + b*sin(c + d*x))^3) - (11*e^3*(e*cos(c + d*x))^(7/2)*(9*a + 4*b*sin(c + d*x)))/(60*b^3*d*(a + b*sin(c + d*x))^2) - (77*e^5*(e*cos(c + d*x))^(3/2)*(15*a^2 - 4*b^2 + 6*a*b*sin(c + d*x)))/(120*b^5*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(11/2)/(a + b*sin(c + d*x))^4, x, 15, -((15*a*(7*a^2 - 6*b^2)*e^(11/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(11/2)*(-a^2 + b^2)^(3/4)*d)) - (15*a*(7*a^2 - 6*b^2)*e^(11/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(11/2)*(-a^2 + b^2)^(3/4)*d) - (5*(21*a^2 - 4*b^2)*e^6*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(8*b^6*d*sqrt(e*cos(c + d*x))) + (15*a^2*(7*a^2 - 6*b^2)*e^6*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (1/2)*(c + d*x), 2))/(16*b^6*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (15*a^2*(7*a^2 - 6*b^2)*e^6*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (1/2)*(c + d*x), 2))/(16*b^6*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(9/2))/(3*b*d*(a + b*sin(c + d*x))^3) - (e^3*(e*cos(c + d*x))^(5/2)*(7*a + 4*b*sin(c + d*x)))/(4*b^3*d*(a + b*sin(c + d*x))^2) - (5*e^5*sqrt(e*cos(c + d*x))*(21*a^2 - 4*b^2 + 14*a*b*sin(c + d*x)))/(8*b^5*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(9/2)/(a + b*sin(c + d*x))^4, x, 15, (7*a*(5*a^2 - 6*b^2)*e^(9/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(9/2)*(-a^2 + b^2)^(5/4)*d) - (7*a*(5*a^2 - 6*b^2)*e^(9/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(9/2)*(-a^2 + b^2)^(5/4)*d) + (7*(5*a^2 - 4*b^2)*e^4*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(8*b^4*(a^2 - b^2)*d*sqrt(cos(c + d*x))) - (7*a^2*(5*a^2 - 6*b^2)*e^5*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b^5*(a^2 - b^2)*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (7*a^2*(5*a^2 - 6*b^2)*e^5*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b^5*(a^2 - b^2)*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(7/2))/(3*b*d*(a + b*sin(c + d*x))^3) + (7*(5*a^2 - 4*b^2)*e^3*(e*cos(c + d*x))^(3/2))/(8*b^3*(a^2 - b^2)*d*(a + b*sin(c + d*x))) - (7*e^3*(e*cos(c + d*x))^(3/2)*(5*a + 4*b*sin(c + d*x)))/(12*b^3*d*(a + b*sin(c + d*x))^2)]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + b*sin(c + d*x))^4, x, 15, (-5*a*(a^2 - 2*b^2)*e^(7/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(7/2)*(-a^2 + b^2)^(7/4)*d) - (5*a*(a^2 - 2*b^2)*e^(7/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(7/2)*(-a^2 + b^2)^(7/4)*d) + (5*(3*a^2 - 4*b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(24*b^4*(a^2 - b^2)*d*sqrt(e*cos(c + d*x))) - (5*a^2*(a^2 - 2*b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b^4*(a^2 - b^2)*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (5*a^2*(a^2 - 2*b^2)*e^4*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b^4*(a^2 - b^2)*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(5/2))/(3*b*d*(a + b*sin(c + d*x))^3) - (5*(3*a^2 - 4*b^2)*e^3*sqrt(e*cos(c + d*x)))/(24*b^3*(a^2 - b^2)*d*(a + b*sin(c + d*x))) + (5*e^3*sqrt(e*cos(c + d*x))*(3*a + 4*b*sin(c + d*x)))/(12*b^3*d*(a + b*sin(c + d*x))^2)]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + b*sin(c + d*x))^4, x, 15, -(a*(a^2 - 6*b^2)*e^(5/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(5/2)*(-a^2 + b^2)^(9/4)*d) + (a*(a^2 - 6*b^2)*e^(5/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(5/2)*(-a^2 + b^2)^(9/4)*d) + ((a^2 + 4*b^2)*e^2*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(8*b^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) - (a^2*(a^2 - 6*b^2)*e^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b^3*(a^2 - b^2)^2*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (a^2*(a^2 - 6*b^2)*e^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b^3*(a^2 - b^2)^2*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) - (e*(e*cos(c + d*x))^(3/2))/(3*b*d*(a + b*sin(c + d*x))^3) + (a*e*(e*cos(c + d*x))^(3/2))/(4*b*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) + ((a^2 + 4*b^2)*e*(e*cos(c + d*x))^(3/2))/(8*b*(a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + b*sin(c + d*x))^4, x, 15, -(a*(a^2 + 6*b^2)*e^(3/2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(3/2)*(-a^2 + b^2)^(11/4)*d) - (a*(a^2 + 6*b^2)*e^(3/2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*b^(3/2)*(-a^2 + b^2)^(11/4)*d) - ((3*a^2 + 4*b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(24*b^2*(a^2 - b^2)^2*d*sqrt(e*cos(c + d*x))) + (a^2*(a^2 + 6*b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b^2*(a^2 - b^2)^2*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (a^2*(a^2 + 6*b^2)*e^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b^2*(a^2 - b^2)^2*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) - (e*sqrt(e*cos(c + d*x)))/(3*b*d*(a + b*sin(c + d*x))^3) + (a*e*sqrt(e*cos(c + d*x)))/(12*b*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) + ((3*a^2 + 4*b^2)*e*sqrt(e*cos(c + d*x)))/(24*b*(a^2 - b^2)^2*d*(a + b*sin(c + d*x)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + b*sin(c + d*x))^4, x, 15, (-5*a*(a^2 + 2*b^2)*sqrt(e)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*sqrt(b)*(-a^2 + b^2)^(13/4)*d) + (5*a*(a^2 + 2*b^2)*sqrt(e)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*sqrt(b)*(-a^2 + b^2)^(13/4)*d) + ((11*a^2 + 4*b^2)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(8*(a^2 - b^2)^3*d*sqrt(cos(c + d*x))) + (5*a^2*(a^2 + 2*b^2)*e*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b*(a^2 - b^2)^3*(b - sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (5*a^2*(a^2 + 2*b^2)*e*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*b*(a^2 - b^2)^3*(b + sqrt(-a^2 + b^2))*d*sqrt(e*cos(c + d*x))) + (b*(e*cos(c + d*x))^(3/2))/(3*(a^2 - b^2)*d*e*(a + b*sin(c + d*x))^3) + (3*a*b*(e*cos(c + d*x))^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*sin(c + d*x))^2) + (b*(11*a^2 + 4*b^2)*(e*cos(c + d*x))^(3/2))/(8*(a^2 - b^2)^3*d*e*(a + b*sin(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^4), x, 15, (7*a*sqrt(b)*(5*a^2 + 6*b^2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*(-a^2 + b^2)^(15/4)*d*sqrt(e)) + (7*a*sqrt(b)*(5*a^2 + 6*b^2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*(-a^2 + b^2)^(15/4)*d*sqrt(e)) - ((57*a^2 + 20*b^2)*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2))/(24*(a^2 - b^2)^3*d*sqrt(e*cos(c + d*x))) + (7*a^2*(5*a^2 + 6*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*(a^2 - b^2)^3*(a^2 - b*(b - sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (7*a^2*(5*a^2 + 6*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*(a^2 - b^2)^3*(a^2 - b*(b + sqrt(-a^2 + b^2)))*d*sqrt(e*cos(c + d*x))) + (b*sqrt(e*cos(c + d*x)))/(3*(a^2 - b^2)*d*e*(a + b*sin(c + d*x))^3) + (11*a*b*sqrt(e*cos(c + d*x)))/(12*(a^2 - b^2)^2*d*e*(a + b*sin(c + d*x))^2) + (b*(57*a^2 + 20*b^2)*sqrt(e*cos(c + d*x)))/(24*(a^2 - b^2)^3*d*e*(a + b*sin(c + d*x)))]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^4), x, 16, (-15*a*b^(3/2)*(7*a^2 + 6*b^2)*atan((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*(-a^2 + b^2)^(17/4)*d*e^(3/2)) + (15*a*b^(3/2)*(7*a^2 + 6*b^2)*atanh((sqrt(b)*sqrt(e*cos(c + d*x)))/((-a^2 + b^2)^(1/4)*sqrt(e))))/(16*(-a^2 + b^2)^(17/4)*d*e^(3/2)) - ((16*a^4 + 151*a^2*b^2 + 28*b^4)*sqrt(e*cos(c + d*x))*Elliptic.E((c + d*x)/2, 2))/(8*(a^2 - b^2)^4*d*e^2*sqrt(cos(c + d*x))) - (15*a^2*b*(7*a^2 + 6*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b - sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*(a^2 - b^2)^4*(b - sqrt(-a^2 + b^2))*d*e*sqrt(e*cos(c + d*x))) - (15*a^2*b*(7*a^2 + 6*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(b + sqrt(-a^2 + b^2)), (c + d*x)/2, 2))/(16*(a^2 - b^2)^4*(b + sqrt(-a^2 + b^2))*d*e*sqrt(e*cos(c + d*x))) + b/(3*(a^2 - b^2)*d*e*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^3) + (13*a*b)/(12*(a^2 - b^2)^2*d*e*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^2) + (b*(89*a^2 + 28*b^2))/(24*(a^2 - b^2)^3*d*e*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))) - (15*a*b*(7*a^2 + 6*b^2) - (16*a^4 + 151*a^2*b^2 + 28*b^4)*sin(c + d*x))/(8*(a^2 - b^2)^4*d*e*sqrt(e*cos(c + d*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^(p/2)*(a+b*sin(e+f*x))^(m/2)=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    #= [(e*cos(c + d*x))^(3/2)*sqrt(a + b*sin(c + d*x)), x, 1, (2*e*AppellF1(3/2, -(1/4), -(1/4), 5/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^(3/2))/(3*b*d*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))]
    @test_int [sqrt(e*cos(c + d*x))*sqrt(a + b*sin(c + d*x)), x, 1, (1/(3*b*d*sqrt(e*cos(c + d*x))))*(2*e*AppellF1(3/2, 1/4, 1/4, 5/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(3/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))]
    @test_int [sqrt(a + b*sin(c + d*x))/sqrt(e*cos(c + d*x)), x, 1, (2*e*AppellF1(3/2, 3/4, 3/4, 5/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(3/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4))/(3*b*d*(e*cos(c + d*x))^(3/2))]
    @test_int [sqrt(a + b*sin(c + d*x))/(e*cos(c + d*x))^(3/2), x, 1, 0]
    @test_int [sqrt(a + b*sin(c + d*x))/(e*cos(c + d*x))^(5/2), x, 2, 0]
    @test_int [sqrt(a + b*sin(c + d*x))/(e*cos(c + d*x))^(7/2), x, 3, 0]
    @test_int [sqrt(a + b*sin(c + d*x))/(e*cos(c + d*x))^(9/2), x, 4, 0] =#

    #= [(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^(3/2), x, 1, (2*e*AppellF1(5/2, -(3/4), -(3/4), 7/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^(5/2))/(5*b*d*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4))]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^(3/2), x, 1, (2*e*AppellF1(5/2, -(1/4), -(1/4), 7/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^(5/2))/(5*b*d*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))]
    @test_int [sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^(3/2), x, 1, (1/(5*b*d*sqrt(e*cos(c + d*x))))*(2*e*AppellF1(5/2, 1/4, 1/4, 7/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(5/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))]
    @test_int [(a + b*sin(c + d*x))^(3/2)/sqrt(e*cos(c + d*x)), x, 1, (1/(5*b*d*(e*cos(c + d*x))^(3/2)))*(2*e*AppellF1(5/2, 3/4, 3/4, 7/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(5/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4))]
    @test_int [(a + b*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(3/2), x, 8, 0]
    @test_int [(a + b*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(5/2), x, 1, 0]
    @test_int [(a + b*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(7/2), x, 3, 0]
    @test_int [(a + b*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(9/2), x, 4, 0]
    @test_int [(a + b*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(11/2), x, 5, 0] =#

    #= [(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^(5/2), x, 1, (2*e*AppellF1(7/2, -(1/4), -(1/4), 9/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^(7/2))/(7*b*d*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))]
    @test_int [sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^(5/2), x, 1, (2*e*AppellF1(7/2, 1/4, 1/4, 9/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(7/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))/(7*b*d*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))^(5/2)/sqrt(e*cos(c + d*x)), x, 1, (2*e*AppellF1(7/2, 3/4, 3/4, 9/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(7/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4))/(7*b*d*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + b*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(3/2), x, 9, 0]
    @test_int [(a + b*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(5/2), x, 8, 0]
    @test_int [(a + b*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(7/2), x, 1, 0]
    @test_int [(a + b*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(9/2), x, 3, 0]
    @test_int [(a + b*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(11/2), x, 4, 0]
    @test_int [(a + b*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(13/2), x, 5, 0] =#

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [1/(sqrt(c*cos(e + f*x))*sqrt(a + b*sin(e + f*x))), x, 2, (2*sqrt(2)*(-a + b)^(1/4)*sqrt(c*cos(e + f*x))*Elliptic.F(asin(((a + b)^(1/4)*sqrt((1 + cos(e + f*x) + sin(e + f*x))/(1 + cos(e + f*x) - sin(e + f*x))))/(-a + b)^(1/4)), -1)*sqrt((a + b*sin(e + f*x))/((a - b)*(1 - sin(e + f*x)))))/((a + b)^(1/4)*c*f*sqrt((1 + cos(e + f*x) + sin(e + f*x))/(1 + cos(e + f*x) - sin(e + f*x)))*sqrt(a + b*sin(e + f*x))), (sqrt(2)*(a - b)^(1/4)*sqrt(c*cos(e + f*x))*Elliptic.F(2*atan(((a + b)^(1/4)*sqrt((1 + cos(e + f*x) + sin(e + f*x))/(1 + cos(e + f*x) - sin(e + f*x))))/(a - b)^(1/4)), 1/2)*sqrt((a + b*sin(e + f*x))/((a - b)*(1 - sin(e + f*x))))*sqrt((a + b*sin(e + f*x))/((a - b)*(1 - cos(f*x)*sin(e) - cos(e)*sin(f*x))*(1 + (sqrt(a + b)*(1 + cos(e + f*x) + sin(e + f*x)))/(sqrt(a - b)*(1 + cos(e + f*x) - sin(e + f*x))))^2))*(1 + (sqrt(a + b)*(1 + cos(e + f*x) + sin(e + f*x)))/(sqrt(a - b)*(1 + cos(e + f*x) - sin(e + f*x)))))/((a + b)^(1/4)*c*f*sqrt((1 + cos(e + f*x) + sin(e + f*x))/(1 + cos(e + f*x) - sin(e + f*x)))*sqrt(a + b*sin(e + f*x))*sqrt((a + b*sin(e + f*x))/((a - b)*(1 - cos(f*x)*sin(e) - cos(e)*sin(f*x)))))]

    #= [(e*cos(c + d*x))^(5/2)/sqrt(a + b*sin(c + d*x)), x, 1, (2*e*AppellF1(1/2, -(3/4), -(3/4), 3/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(e*cos(c + d*x))^(3/2)*sqrt(a + b*sin(c + d*x)))/(b*d*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4))]
    @test_int [(e*cos(c + d*x))^(3/2)/sqrt(a + b*sin(c + d*x)), x, 1, (2*e*AppellF1(1/2, -(1/4), -(1/4), 3/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*sqrt(e*cos(c + d*x))*sqrt(a + b*sin(c + d*x)))/(b*d*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))]
    @test_int [sqrt(e*cos(c + d*x))/sqrt(a + b*sin(c + d*x)), x, 1, (2*e*AppellF1(1/2, 1/4, 1/4, 3/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*sqrt(a + b*sin(c + d*x))*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))/(b*d*sqrt(e*cos(c + d*x)))]
    @test_int [1/(sqrt(e*cos(c + d*x))*sqrt(a + b*sin(c + d*x))), x, 1, 0]
    @test_int [1/((e*cos(c + d*x))^(3/2)*sqrt(a + b*sin(c + d*x))), x, 2, 0]
    @test_int [1/((e*cos(c + d*x))^(5/2)*sqrt(a + b*sin(c + d*x))), x, 3, 0]
    @test_int [1/((e*cos(c + d*x))^(7/2)*sqrt(a + b*sin(c + d*x))), x, 4, 0] =#

    #= [(e*cos(c + d*x))^(7/2)/(a + b*sin(c + d*x))^(3/2), x, 1, -((2*e*AppellF1(-(1/2), -(5/4), -(5/4), 1/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(e*cos(c + d*x))^(5/2))/(b*d*sqrt(a + b*sin(c + d*x))*(1 - (a + b*sin(c + d*x))/(a - b))^(5/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(5/4)))]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + b*sin(c + d*x))^(3/2), x, 1, -((2*e*AppellF1(-(1/2), -(3/4), -(3/4), 1/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(e*cos(c + d*x))^(3/2))/(b*d*sqrt(a + b*sin(c + d*x))*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + b*sin(c + d*x))^(3/2), x, 1, -((2*e*AppellF1(-(1/2), -(1/4), -(1/4), 1/2, (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*sqrt(e*cos(c + d*x)))/(b*d*sqrt(a + b*sin(c + d*x))*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4)))]
    @test_int [sqrt(e*cos(c + d*x))/(a + b*sin(c + d*x))^(3/2), x, 1, -((2*2^(3/4)*e*HypergeometricFunctions._₂F₁(1/4, 3/4, 7/4, ((a - b)*(1 - sin(c + d*x)))/(2*(a + b*sin(c + d*x))))*(1 - sin(c + d*x))*(((a + b)*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^(1/4))/(3*(a + b)*d*sqrt(e*cos(c + d*x))*sqrt(a + b*sin(c + d*x))))]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^(3/2)), x, 2, 0]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^(3/2)), x, 3, 0]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^(3/2)), x, 4, 0]
    @test_int [1/((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^(3/2)), x, 5, 0] =#

    #= [(e*cos(c + d*x))^(9/2)/(a + b*sin(c + d*x))^(5/2), x, 1, -((2*e*AppellF1(-(3/2), -(7/4), -(7/4), -(1/2), (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(e*cos(c + d*x))^(7/2))/(3*b*d*(a + b*sin(c + d*x))^(3/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(7/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(7/4)))]
    @test_int [(e*cos(c + d*x))^(7/2)/(a + b*sin(c + d*x))^(5/2), x, 1, -((2*e*AppellF1(-(3/2), -(5/4), -(5/4), -(1/2), (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(e*cos(c + d*x))^(5/2))/(3*b*d*(a + b*sin(c + d*x))^(3/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(5/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(5/4)))]
    @test_int [(e*cos(c + d*x))^(5/2)/(a + b*sin(c + d*x))^(5/2), x, 1, -((2*e*AppellF1(-(3/2), -(3/4), -(3/4), -(1/2), (a + b*sin(c + d*x))/(a + b), (a + b*sin(c + d*x))/(a - b))*(e*cos(c + d*x))^(3/2))/(3*b*d*(a + b*sin(c + d*x))^(3/2)*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4)))]
    @test_int [(e*cos(c + d*x))^(3/2)/(a + b*sin(c + d*x))^(5/2), x, 1, 0]
    @test_int [sqrt(e*cos(c + d*x))/(a + b*sin(c + d*x))^(5/2), x, 2, 0]
    @test_int [1/(sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^(5/2)), x, 3, 0]
    @test_int [1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^(5/2)), x, 4, 0]
    @test_int [1/((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^(5/2)), x, 5, 0] =#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^p*(a+b*sin(e+f*x))^m*when*p*symbolic=#

    #= [(e*cos(c + d*x))^p*(a + b*sin(c + d*x))^8, x, 10, -((a*b*(9 + m)*(3*b^6*(5053 + 4443*m + 1163*m^2 + 93*m^3) + a^2*b^4*(101978 + 91503*m + 28373*m^2 + 3753*m^3 + 185*m^4) + a^4*b^2*(95912 + 88506*m + 33953*m^2 + 6723*m^3 + 671*m^4 + 27*m^5) + a^6*(12176 + 11772*m + 6168*m^2 + 1809*m^3 + 303*m^4 + 27*m^5 + m^6))*(e*cos(c + d*x))^(1 + m))/(d*e*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(2 + 3*m + m^2))) + (1/(d*e*(6 + m)*(8 + m)*(8 + 6*m + m^2)))*((105*b^8 + 420*a^2*b^6*(8 + m) + 210*a^4*b^4*(48 + 14*m + m^2) + 28*a^6*b^2*(192 + 104*m + 18*m^2 + m^3) + a^8*(384 + 400*m + 140*m^2 + 20*m^3 + m^4))*(e*cos(c + d*x))^(1 + m)*(cos(c + d*x)^2)^((1/2)*(-1 - m))*HypergeometricFunctions._₂F₁(1/2, (1 - m)/2, 3/2, sin(c + d*x)^2)*sin(c + d*x)) - (1/(d*e*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(6 + 5*m + m^2)))*(b*(105*b^6*(105 + 71*m + 15*m^2 + m^3) + 3*a^2*b^4*(75798 + 51665*m + 12355*m^2 + 1255*m^3 + 47*m^4) + a^4*b^2*(367992 + 253870*m + 72195*m^2 + 10705*m^3 + 813*m^4 + 25*m^5) + a^6*(69264 + 48860*m + 18424*m^2 + 4025*m^3 + 511*m^4 + 35*m^5 + m^6))*(e*cos(c + d*x))^(1 + m)*(a + b*sin(c + d*x))) - (a*b*(11 + m)*(3*b^4*(783 + 308*m + 29*m^2) + 2*a^2*b^2*(3852 + 1529*m + 208*m^2 + 11*m^3) + a^4*(2232 + 902*m + 203*m^2 + 22*m^3 + m^4))*(e*cos(c + d*x))^(1 + m)*(a + b*sin(c + d*x))^2)/(d*e*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (b*(35*b^4*(35 + 12*m + m^2) + 2*a^2*b^2*(6068 + 2091*m + 232*m^2 + 9*m^3) + a^4*(5944 + 2070*m + 355*m^2 + 30*m^3 + m^4))*(e*cos(c + d*x))^(1 + m)*(a + b*sin(c + d*x))^3)/(d*e*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (a*b*(13 + m)*(b^2*(83 + 13*m) + a^2*(82 + 13*m + m^2))*(e*cos(c + d*x))^(1 + m)*(a + b*sin(c + d*x))^4)/(d*e*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (b*(7*b^2*(7 + m) + a^2*(146 + 21*m + m^2))*(e*cos(c + d*x))^(1 + m)*(a + b*sin(c + d*x))^5)/(d*e*(6 + m)*(7 + m)*(8 + m)) - (a*b*(15 + m)*(e*cos(c + d*x))^(1 + m)*(a + b*sin(c + d*x))^6)/(d*e*(7 + m)*(8 + m)) - (b*(e*cos(c + d*x))^(1 + m)*(a + b*sin(c + d*x))^7)/(d*e*(8 + m))] =#
    @test_int [(e*cos(c + d*x))^p*(a + b*sin(c + d*x))^3, x, 4, If(13>=8, -((b*(2*b^2*(2 + p) + a^2*(11 + 6*p + p^2))*(e*cos(c + d*x))^(1 + p))/(d*e*(1 + p)*(2 + p)*(3 + p))) - (a*(3*b^2 + a^2*(2 + p))*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁(1/2, (1 + p)/2, (3 + p)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*e*(1 + p)*(2 + p)*sqrt(sin(c + d*x)^2)) - (a*b*(5 + p)*(e*cos(c + d*x))^(1 + p)*(a + b*sin(c + d*x)))/(d*e*(2 + p)*(3 + p)) - (b*(e*cos(c + d*x))^(1 + p)*(a + b*sin(c + d*x))^2)/(d*e*(3 + p)), -((b*(2*b^2*(2 + p) + a^2*(11 + 6*p + p^2))*(e*cos(c + d*x))^(1 + p))/(d*e*(3 + p)*(2 + 3*p + p^2))) - (a*(3*b^2 + a^2*(2 + p))*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁(1/2, (1 + p)/2, (3 + p)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*e*(1 + p)*(2 + p)*sqrt(sin(c + d*x)^2)) - (a*b*(5 + p)*(e*cos(c + d*x))^(1 + p)*(a + b*sin(c + d*x)))/(d*e*(2 + p)*(3 + p)) - (b*(e*cos(c + d*x))^(1 + p)*(a + b*sin(c + d*x))^2)/(d*e*(3 + p)))]
    @test_int [(e*cos(c + d*x))^p*(a + b*sin(c + d*x))^2, x, 3, -((a*b*(3 + p)*(e*cos(c + d*x))^(1 + p))/(d*e*(1 + p)*(2 + p))) - ((b^2 + a^2*(2 + p))*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁(1/2, (1 + p)/2, (3 + p)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*e*(1 + p)*(2 + p)*sqrt(sin(c + d*x)^2)) - (b*(e*cos(c + d*x))^(1 + p)*(a + b*sin(c + d*x)))/(d*e*(2 + p))]
    @test_int [(e*cos(c + d*x))^p*(a + b*sin(c + d*x)), x, 2, -((b*(e*cos(c + d*x))^(1 + p))/(d*e*(1 + p))) - (a*(e*cos(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁(1/2, (1 + p)/2, (3 + p)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*e*(1 + p)*sqrt(sin(c + d*x)^2))]
    @test_int [(e*cos(c + d*x))^p/(a + b*sin(c + d*x)), x, 1, -((e*AppellF1(1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*sin(c + d*x)), (a - b)/(a + b*sin(c + d*x)))*(e*cos(c + d*x))^(-1 + p)*(-((b*(1 - sin(c + d*x)))/(a + b*sin(c + d*x))))^((1 - p)/2)*((b*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((1 - p)/2))/(b*d*(1 - p)))]
    @test_int [(e*cos(c + d*x))^p/(a + b*sin(c + d*x))^2, x, 1, -((e*AppellF1(2 - p, (1 - p)/2, (1 - p)/2, 3 - p, (a + b)/(a + b*sin(c + d*x)), (a - b)/(a + b*sin(c + d*x)))*(e*cos(c + d*x))^(-1 + p)*(-((b*(1 - sin(c + d*x)))/(a + b*sin(c + d*x))))^((1 - p)/2)*((b*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((1 - p)/2))/(b*d*(2 - p)*(a + b*sin(c + d*x))))]
    @test_int [(e*cos(c + d*x))^p/(a + b*sin(c + d*x))^3, x, 1, -((e*AppellF1(3 - p, (1 - p)/2, (1 - p)/2, 4 - p, (a + b)/(a + b*sin(c + d*x)), (a - b)/(a + b*sin(c + d*x)))*(e*cos(c + d*x))^(-1 + p)*(-((b*(1 - sin(c + d*x)))/(a + b*sin(c + d*x))))^((1 - p)/2)*((b*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((1 - p)/2))/(b*d*(3 - p)*(a + b*sin(c + d*x))^2))]
    @test_int [(e*cos(c + d*x))^p/(a + b*sin(c + d*x))^8, x, 1, -((e*AppellF1(8 - p, (1 - p)/2, (1 - p)/2, 9 - p, (a + b)/(a + b*sin(c + d*x)), (a - b)/(a + b*sin(c + d*x)))*(e*cos(c + d*x))^(-1 + p)*(-((b*(1 - sin(c + d*x)))/(a + b*sin(c + d*x))))^((1 - p)/2)*((b*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((1 - p)/2))/(b*d*(8 - p)*(a + b*sin(c + d*x))^7))]

    @test_int [(e*cos(c + d*x))^p*(a + b*sin(c + d*x))^(5/2), x, 2, (2*e*AppellF1(7/2, (1 - p)/2, (1 - p)/2, 9/2, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(-1 + p)*(a + b*sin(c + d*x))^(7/2)*(1 - (a + b*sin(c + d*x))/(a - b))^((1 - p)/2)*(1 - (a + b*sin(c + d*x))/(a + b))^((1 - p)/2))/(7*b*d)]
    @test_int [(e*cos(c + d*x))^p*(a + b*sin(c + d*x))^(3/2), x, 2, (2*e*AppellF1(5/2, (1 - p)/2, (1 - p)/2, 7/2, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(-1 + p)*(a + b*sin(c + d*x))^(5/2)*(1 - (a + b*sin(c + d*x))/(a - b))^((1 - p)/2)*(1 - (a + b*sin(c + d*x))/(a + b))^((1 - p)/2))/(5*b*d)]
    @test_int [(e*cos(c + d*x))^p*sqrt(a + b*sin(c + d*x)), x, 2, (2*e*AppellF1(3/2, (1 - p)/2, (1 - p)/2, 5/2, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(-1 + p)*(a + b*sin(c + d*x))^(3/2)*(1 - (a + b*sin(c + d*x))/(a - b))^((1 - p)/2)*(1 - (a + b*sin(c + d*x))/(a + b))^((1 - p)/2))/(3*b*d)]
    @test_int [(e*cos(c + d*x))^p/sqrt(a + b*sin(c + d*x)), x, 2, (2*e*AppellF1(1/2, (1 - p)/2, (1 - p)/2, 3/2, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(-1 + p)*sqrt(a + b*sin(c + d*x))*(1 - (a + b*sin(c + d*x))/(a - b))^((1 - p)/2)*(1 - (a + b*sin(c + d*x))/(a + b))^((1 - p)/2))/(b*d)]
    @test_int [(e*cos(c + d*x))^p/(a + b*sin(c + d*x))^(3/2), x, 2, -((2*e*AppellF1(-(1/2), (1 - p)/2, (1 - p)/2, 1/2, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(-1 + p)*(1 - (a + b*sin(c + d*x))/(a - b))^((1 - p)/2)*(1 - (a + b*sin(c + d*x))/(a + b))^((1 - p)/2))/(b*d*sqrt(a + b*sin(c + d*x))))]
    @test_int [(e*cos(c + d*x))^p/(a + b*sin(c + d*x))^(5/2), x, 2, -((2*e*AppellF1(-(3/2), (1 - p)/2, (1 - p)/2, -(1/2), (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(-1 + p)*(1 - (a + b*sin(c + d*x))/(a - b))^((1 - p)/2)*(1 - (a + b*sin(c + d*x))/(a + b))^((1 - p)/2))/(3*b*d*(a + b*sin(c + d*x))^(3/2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^p*(a+b*sin(e+f*x))^m*when*m*symbolic=#

    @test_int [(e*cos(c + d*x))^p*(a + b*sin(c + d*x))^m, x, 2, (e*AppellF1(1 + m, (1 - p)/2, (1 - p)/2, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(-1 + p)*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^((1 - p)/2)*(1 - (a + b*sin(c + d*x))/(a + b))^((1 - p)/2))/(b*d*(1 + m))]

    @test_int [cos(c + d*x)^7*(a + b*sin(c + d*x))^m, x, 3, -(((a^2 - b^2)^3*(a + b*sin(c + d*x))^(1 + m))/(b^7*d*(1 + m))) + (6*a*(a^2 - b^2)^2*(a + b*sin(c + d*x))^(2 + m))/(b^7*d*(2 + m)) - (3*(5*a^4 - 6*a^2*b^2 + b^4)*(a + b*sin(c + d*x))^(3 + m))/(b^7*d*(3 + m)) + (4*a*(5*a^2 - 3*b^2)*(a + b*sin(c + d*x))^(4 + m))/(b^7*d*(4 + m)) - (3*(5*a^2 - b^2)*(a + b*sin(c + d*x))^(5 + m))/(b^7*d*(5 + m)) + (6*a*(a + b*sin(c + d*x))^(6 + m))/(b^7*d*(6 + m)) - (a + b*sin(c + d*x))^(7 + m)/(b^7*d*(7 + m))]
    @test_int [cos(c + d*x)^5*(a + b*sin(c + d*x))^m, x, 3, ((a^2 - b^2)^2*(a + b*sin(c + d*x))^(1 + m))/(b^5*d*(1 + m)) - (4*a*(a^2 - b^2)*(a + b*sin(c + d*x))^(2 + m))/(b^5*d*(2 + m)) + (2*(3*a^2 - b^2)*(a + b*sin(c + d*x))^(3 + m))/(b^5*d*(3 + m)) - (4*a*(a + b*sin(c + d*x))^(4 + m))/(b^5*d*(4 + m)) + (a + b*sin(c + d*x))^(5 + m)/(b^5*d*(5 + m))]
    @test_int [cos(c + d*x)^3*(a + b*sin(c + d*x))^m, x, 3, -(((a^2 - b^2)*(a + b*sin(c + d*x))^(1 + m))/(b^3*d*(1 + m))) + (2*a*(a + b*sin(c + d*x))^(2 + m))/(b^3*d*(2 + m)) - (a + b*sin(c + d*x))^(3 + m)/(b^3*d*(3 + m))]
    @test_int [cos(c + d*x)^1*(a + b*sin(c + d*x))^m, x, 2, (a + b*sin(c + d*x))^(1 + m)/(b*d*(1 + m))]
    @test_int [sec(c + d*x)^1*(a + b*sin(c + d*x))^m, x, 5, -((HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(1 + m))/(2*(a - b)*d*(1 + m))) + (HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*sin(c + d*x))/(a + b))*(a + b*sin(c + d*x))^(1 + m))/(2*(a + b)*d*(1 + m))]
    @test_int [sec(c + d*x)^3*(a + b*sin(c + d*x))^m, x, 6, -(((a - b*(1 - m))*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(1 + m))/(4*(a - b)^2*d*(1 + m))) + ((a + b - b*m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*sin(c + d*x))/(a + b))*(a + b*sin(c + d*x))^(1 + m))/(4*(a + b)^2*d*(1 + m)) - (sec(c + d*x)^2*(b - a*sin(c + d*x))*(a + b*sin(c + d*x))^(1 + m))/(2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^5*(a + b*sin(c + d*x))^m, x, 7, -(((3*a^2 - 3*a*b*(2 - m) + b^2*(3 - 4*m + m^2))*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*sin(c + d*x))/(a - b))*(a + b*sin(c + d*x))^(1 + m))/(16*(a - b)^3*d*(1 + m))) + ((3*a^2 + 3*a*b*(2 - m) + b^2*(3 - 4*m + m^2))*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (a + b*sin(c + d*x))/(a + b))*(a + b*sin(c + d*x))^(1 + m))/(16*(a + b)^3*d*(1 + m)) - (sec(c + d*x)^4*(b - a*sin(c + d*x))*(a + b*sin(c + d*x))^(1 + m))/(4*(a^2 - b^2)*d) + (sec(c + d*x)^2*(a + b*sin(c + d*x))^(1 + m)*(b*(b^2*(3 - m) - a^2*(1 + m)) + a*(3*a^2 - b^2*(5 - 2*m))*sin(c + d*x)))/(8*(a^2 - b^2)^2*d)]

    @test_int [cos(c + d*x)^4*(a + b*sin(c + d*x))^m, x, 2, (AppellF1(1 + m, -(3/2), -(3/2), 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*cos(c + d*x)^3*(a + b*sin(c + d*x))^(1 + m))/(b*d*(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(3/2)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/2))]
    @test_int [cos(c + d*x)^2*(a + b*sin(c + d*x))^m, x, 2, (AppellF1(1 + m, -(1/2), -(1/2), 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^(1 + m))/(b*d*(1 + m)*sqrt(1 - (a + b*sin(c + d*x))/(a - b))*sqrt(1 - (a + b*sin(c + d*x))/(a + b)))]
    @test_int [sec(c + d*x)^2*(a + b*sin(c + d*x))^m, x, 2, (AppellF1(1 + m, 3/2, 3/2, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*sec(c + d*x)^3*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(3/2)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/2))/(b*d*(1 + m))]
    @test_int [sec(c + d*x)^4*(a + b*sin(c + d*x))^m, x, 2, (AppellF1(1 + m, 5/2, 5/2, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*sec(c + d*x)^5*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(5/2)*(1 - (a + b*sin(c + d*x))/(a + b))^(5/2))/(b*d*(1 + m))]

    @test_int [(e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^m, x, 2, (e*AppellF1(1 + m, -(3/4), -(3/4), 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^(1 + m))/(b*d*(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4))]
    @test_int [(e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^m, x, 2, (e*AppellF1(1 + m, -(1/4), -(1/4), 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*sqrt(e*cos(c + d*x))*(a + b*sin(c + d*x))^(1 + m))/(b*d*(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))]
    @test_int [(e*cos(c + d*x))^(1/2)*(a + b*sin(c + d*x))^m, x, 2, (e*AppellF1(1 + m, 1/4, 1/4, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(1/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(1/4))/(b*d*(1 + m)*sqrt(e*cos(c + d*x)))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(1/2), x, 2, (e*AppellF1(1 + m, 3/4, 3/4, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(3/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(3/4))/(b*d*(1 + m)*(e*cos(c + d*x))^(3/2))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(3/2), x, 2, (e*AppellF1(1 + m, 5/4, 5/4, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(5/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(5/4))/(b*d*(1 + m)*(e*cos(c + d*x))^(5/2))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(5/2), x, 2, (e*AppellF1(1 + m, 7/4, 7/4, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(7/4)*(1 - (a + b*sin(c + d*x))/(a + b))^(7/4))/(b*d*(1 + m)*(e*cos(c + d*x))^(7/2))]

    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 4), x, 9, If(13>=8, -(((e*cos(c + d*x))^(-3 - m)*(a + b*sin(c + d*x))^(1 + m))/((a - b)*d*e*(3 + m))) + (2*b*(e*cos(c + d*x))^(-1 - m)*(a + b*sin(c + d*x))^(1 + m))/((a - b)^2*d*e^3*(1 + m)*(3 + m)) + (a*(e*cos(c + d*x))^(-3 - m)*(1 + sin(c + d*x))*(a + b*sin(c + d*x))^(1 + m))/((a^2 - b^2)*d*e*(3 + m)) + (a*(3*b + a*(2 + m))*(e*cos(c + d*x))^(-3 - m)*(1 - sin(c + d*x))*(1 + sin(c + d*x))*(a + b*sin(c + d*x))^(1 + m))/((a - b)*(a + b)^2*d*e*(1 + m)*(3 + m)) - (2^(3/2 - m/2)*a*b*(e*cos(c + d*x))^(-1 - m)*HypergeometricFunctions._₂F₁((1/2)*(-1 - m), (1 + m)/2, (1 - m)/2, ((a - b)*(1 - sin(c + d*x)))/(2*(a + b*sin(c + d*x))))*(((a + b)*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((1 + m)/2)*(a + b*sin(c + d*x))^(1 + m))/((a - b)^2*(a + b)*d*e^3*(1 + m)*(3 + m)) - (2^(-(1/2) - m/2)*a*(2*a*b - b^2 + a^2*(2 + m))*(e*cos(c + d*x))^(-3 - m)*HypergeometricFunctions._₂F₁((1 - m)/2, (3 + m)/2, (3 - m)/2, ((a - b)*(1 - sin(c + d*x)))/(2*(a + b*sin(c + d*x))))*(1 - sin(c + d*x))^2*(((a + b)*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((3 + m)/2)*(a + b*sin(c + d*x))^(1 + m))/((a - b)*(a + b)^3*d*e*(1 - m)*(3 + m)), -(((e*cos(c + d*x))^(-3 - m)*(a + b*sin(c + d*x))^(1 + m))/((a - b)*d*e*(3 + m))) + (2*b*(e*cos(c + d*x))^(-1 - m)*(a + b*sin(c + d*x))^(1 + m))/((a - b)^2*d*e^3*(1 + m)*(3 + m)) + (a*(e*cos(c + d*x))^(-3 - m)*(1 + sin(c + d*x))*(a + b*sin(c + d*x))^(1 + m))/((a^2 - b^2)*d*e*(3 + m)) + (a*(3*b + a*(2 + m))*(e*cos(c + d*x))^(-3 - m)*(1 - sin(c + d*x))*(1 + sin(c + d*x))*(a + b*sin(c + d*x))^(1 + m))/((a - b)*(a + b)^2*d*e*(1 + m)*(3 + m)) - (2^(3/2 - m/2)*a*b*(e*cos(c + d*x))^(-1 - m)*HypergeometricFunctions._₂F₁((1/2)*(-1 - m), (1 + m)/2, (1 - m)/2, ((a - b)*(1 - sin(c + d*x)))/(2*(a + b*sin(c + d*x))))*(((a + b)*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((1 + m)/2)*(a + b*sin(c + d*x))^(1 + m))/((a - b)^2*(a + b)*d*e^3*(3 + 4*m + m^2)) - (2^(-(1/2) - m/2)*a*(2*a*b - b^2 + a^2*(2 + m))*(e*cos(c + d*x))^(-3 - m)*HypergeometricFunctions._₂F₁((1 - m)/2, (3 + m)/2, (3 - m)/2, ((a - b)*(1 - sin(c + d*x)))/(2*(a + b*sin(c + d*x))))*(1 - sin(c + d*x))^2*(((a + b)*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((3 + m)/2)*(a + b*sin(c + d*x))^(1 + m))/((a - b)*(a + b)^3*d*e*(1 - m)*(3 + m)))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 3), x, -5, (sec(c + d*x)^4*(-1 + sin(c + d*x))*(1 + sin(c + d*x))*(a + b*sin(c + d*x))^(1 + m))/((e*cos(c + d*x))^m*((a - b)*d*e^3*(2 + m))) + ((-2*b + a*(2 + m))*sec(c + d*x)^4*(-1 + sin(c + d*x))*(1 + sin(c + d*x))^2*(a + b*sin(c + d*x))^(1 + m))/((e*cos(c + d*x))^m*((a - b)^2*d*e^3*m*(2 + m))) - ((-b^2 + a^2*(1 + m))*HypergeometricFunctions._₂F₁(m/2, 1 + m, 2 + m, -((2*(a + b*sin(c + d*x)))/((a - b)*(-1 + sin(c + d*x)))))*sec(c + d*x)^4*(1 + sin(c + d*x))^3*(((a + b)*(1 + sin(c + d*x)))/((a - b)*(-1 + sin(c + d*x))))^((1/2)*(-2 + m))*(a + b*sin(c + d*x))^(1 + m))/((e*cos(c + d*x))^m*((a - b)^3*d*e^3*m*(1 + m)))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 2), x, 3, -(((e*cos(c + d*x))^(-1 - m)*(a + b*sin(c + d*x))^(1 + m))/((a - b)*d*e*(1 + m))) + (2^(1/2 - m/2)*a*(e*cos(c + d*x))^(-1 - m)*HypergeometricFunctions._₂F₁((1/2)*(-1 - m), (1 + m)/2, (1 - m)/2, ((a - b)*(1 - sin(c + d*x)))/(2*(a + b*sin(c + d*x))))*(((a + b)*(1 + sin(c + d*x)))/(a + b*sin(c + d*x)))^((1 + m)/2)*(a + b*sin(c + d*x))^(1 + m))/((a^2 - b^2)*d*e*(1 + m))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 1), x, 1, (e*(e*cos(c + d*x))^(-2 - m)*HypergeometricFunctions._₂F₁(1 + m, (2 + m)/2, 2 + m, (2*(a + b*sin(c + d*x)))/((a + b)*(1 + sin(c + d*x))))*(1 - sin(c + d*x))*(-(((a - b)*(1 - sin(c + d*x)))/((a + b)*(1 + sin(c + d*x)))))^(m/2)*(a + b*sin(c + d*x))^(1 + m))/((a + b)*d*(1 + m))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m + 0), x, 2, (e*AppellF1(1 + m, (1 + m)/2, (1 + m)/2, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(-1 - m)*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^((1 + m)/2)*(1 - (a + b*sin(c + d*x))/(a + b))^((1 + m)/2))/(b*d*(1 + m))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m - 1), x, 2, (e*AppellF1(1 + m, m/2, m/2, 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^(m/2)*(1 - (a + b*sin(c + d*x))/(a + b))^(m/2))/((e*cos(c + d*x))^m*(b*d*(1 + m)))]
    @test_int [(a + b*sin(c + d*x))^m/(e*cos(c + d*x))^(m - 2), x, 2, (e*AppellF1(1 + m, (1/2)*(-1 + m), (1/2)*(-1 + m), 2 + m, (a + b*sin(c + d*x))/(a - b), (a + b*sin(c + d*x))/(a + b))*(e*cos(c + d*x))^(1 - m)*(a + b*sin(c + d*x))^(1 + m)*(1 - (a + b*sin(c + d*x))/(a - b))^((1/2)*(-1 + m))*(1 - (a + b*sin(c + d*x))/(a + b))^((1/2)*(-1 + m)))/(b*d*(1 + m))]

    #= ::Title:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+b*sin(e+f*x))^m=#

    #= ::Section:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+b*sin(e+f*x))^m*when*a^2-b^2=0=#

    #= ::Section:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+b*sin(e+f*x))^m=#
    end

max_stars_repo_path: test/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.jl