text:
@testset "5.5.1 u (a+b arcsec(c x))^n" begin
    (a, b, c, d, e, f, m, x, ) = @variables a b c d e f m x

    #= ::Package:: =#

    #= ::Title:: =#
    #=Integrands*of*the*form*u*(a+b*asec(c*x))^n=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*asec(c*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*asec(c*x))^n=#

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

    @test_int [x^6*(a + b*asec(c*x)), x, 7, -((5*b*sqrt(1 - 1/(c^2*x^2))*x^2)/(112*c^5)) - (5*b*sqrt(1 - 1/(c^2*x^2))*x^4)/(168*c^3) - (b*sqrt(1 - 1/(c^2*x^2))*x^6)/(42*c) + (1/7)*x^7*(a + b*asec(c*x)) - (5*b*atanh(sqrt(1 - 1/(c^2*x^2))))/(112*c^7)]
    @test_int [x^5*(a + b*asec(c*x)), x, 4, -((4*b*sqrt(1 - 1/(c^2*x^2))*x)/(45*c^5)) - (2*b*sqrt(1 - 1/(c^2*x^2))*x^3)/(45*c^3) - (b*sqrt(1 - 1/(c^2*x^2))*x^5)/(30*c) + (1/6)*x^6*(a + b*asec(c*x))]
    @test_int [x^4*(a + b*asec(c*x)), x, 6, -((3*b*sqrt(1 - 1/(c^2*x^2))*x^2)/(40*c^3)) - (b*sqrt(1 - 1/(c^2*x^2))*x^4)/(20*c) + (1/5)*x^5*(a + b*asec(c*x)) - (3*b*atanh(sqrt(1 - 1/(c^2*x^2))))/(40*c^5)]
    @test_int [x^3*(a + b*asec(c*x)), x, 3, -((b*sqrt(1 - 1/(c^2*x^2))*x)/(6*c^3)) - (b*sqrt(1 - 1/(c^2*x^2))*x^3)/(12*c) + (1/4)*x^4*(a + b*asec(c*x))]
    @test_int [x^2*(a + b*asec(c*x)), x, 5, -((b*sqrt(1 - 1/(c^2*x^2))*x^2)/(6*c)) + (1/3)*x^3*(a + b*asec(c*x)) - (b*atanh(sqrt(1 - 1/(c^2*x^2))))/(6*c^3)]
    @test_int [x^1*(a + b*asec(c*x)), x, 2, -((b*sqrt(1 - 1/(c^2*x^2))*x)/(2*c)) + (1/2)*x^2*(a + b*asec(c*x))]
    @test_int [x^0*(a + b*asec(c*x)), x, 5, a*x + b*x*asec(c*x) - (b*atanh(sqrt(1 - 1/(c^2*x^2))))/c]
    @test_int [(a + b*asec(c*x))/x^1, x, 6, (im*(a + b*asec(c*x))^2)/(2*b) - (a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))) + (1/2)*im*b*PolyLog(2, -ℯ^(2*im*asec(c*x)))]
    @test_int [(a + b*asec(c*x))/x^2, x, 2, b*c*sqrt(1 - 1/(c^2*x^2)) - (a + b*asec(c*x))/x]
    @test_int [(a + b*asec(c*x))/x^3, x, 4, (b*c*sqrt(1 - 1/(c^2*x^2)))/(4*x) - (a + b*asec(c*x))/(2*x^2) - (1/4)*b*c^2*acsc(c*x)]
    @test_int [(a + b*asec(c*x))/x^4, x, 4, (1/3)*b*c^3*sqrt(1 - 1/(c^2*x^2)) - (1/9)*b*c^3*(1 - 1/(c^2*x^2))^(3/2) - (a + b*asec(c*x))/(3*x^3)]
    @test_int [(a + b*asec(c*x))/x^5, x, 5, (b*c*sqrt(1 - 1/(c^2*x^2)))/(16*x^3) + (3*b*c^3*sqrt(1 - 1/(c^2*x^2)))/(32*x) - (a + b*asec(c*x))/(4*x^4) - (3/32)*b*c^4*acsc(c*x)]
    @test_int [(a + b*asec(c*x))/x^6, x, 4, (1/5)*b*c^5*sqrt(1 - 1/(c^2*x^2)) - (2/15)*b*c^5*(1 - 1/(c^2*x^2))^(3/2) + (1/25)*b*c^5*(1 - 1/(c^2*x^2))^(5/2) - (a + b*asec(c*x))/(5*x^5)]
    @test_int [(a + b*asec(c*x))/x^7, x, 6, (b*c*sqrt(1 - 1/(c^2*x^2)))/(36*x^5) + (5*b*c^3*sqrt(1 - 1/(c^2*x^2)))/(144*x^3) + (5*b*c^5*sqrt(1 - 1/(c^2*x^2)))/(96*x) - (a + b*asec(c*x))/(6*x^6) - (5/96)*b*c^6*acsc(c*x)]

    @test_int [x^3*(a + b*asec(c*x))^2, x, 5, (b^2*x^2)/(12*c^2) - (b*sqrt(1 - 1/(c^2*x^2))*x*(a + b*asec(c*x)))/(3*c^3) - (b*sqrt(1 - 1/(c^2*x^2))*x^3*(a + b*asec(c*x)))/(6*c) + (1/4)*x^4*(a + b*asec(c*x))^2 + (b^2*log(x))/(3*c^4)]
    @test_int [x^2*(a + b*asec(c*x))^2, x, 8, (b^2*x)/(3*c^2) - (b*sqrt(1 - 1/(c^2*x^2))*x^2*(a + b*asec(c*x)))/(3*c) + (1/3)*x^3*(a + b*asec(c*x))^2 + (2*im*b*(a + b*asec(c*x))*atan(ℯ^(im*asec(c*x))))/(3*c^3) - (im*b^2*PolyLog(2, (-im)*ℯ^(im*asec(c*x))))/(3*c^3) + (im*b^2*PolyLog(2, im*ℯ^(im*asec(c*x))))/(3*c^3)]
    @test_int [x^1*(a + b*asec(c*x))^2, x, 4, -((b*sqrt(1 - 1/(c^2*x^2))*x*(a + b*asec(c*x)))/c) + (1/2)*x^2*(a + b*asec(c*x))^2 + (b^2*log(x))/c^2]
    @test_int [x^0*(a + b*asec(c*x))^2, x, 7, x*(a + b*asec(c*x))^2 + (4*im*b*(a + b*asec(c*x))*atan(ℯ^(im*asec(c*x))))/c - (2*im*b^2*PolyLog(2, (-im)*ℯ^(im*asec(c*x))))/c + (2*im*b^2*PolyLog(2, im*ℯ^(im*asec(c*x))))/c]
    @test_int [(a + b*asec(c*x))^2/x^1, x, 6, (im*(a + b*asec(c*x))^3)/(3*b) - (a + b*asec(c*x))^2*log(1 + ℯ^(2*im*asec(c*x))) + im*b*(a + b*asec(c*x))*PolyLog(2, -ℯ^(2*im*asec(c*x))) - (1/2)*b^2*PolyLog(3, -ℯ^(2*im*asec(c*x)))]
    @test_int [(a + b*asec(c*x))^2/x^2, x, 4, (2*b^2)/x + 2*b*c*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x)) - (a + b*asec(c*x))^2/x]
    @test_int [(a + b*asec(c*x))^2/x^3, x, 4, b^2/(4*x^2) - (1/2)*a*b*c^2*asec(c*x) - (1/4)*b^2*c^2*asec(c*x)^2 + (b*c*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x)))/(2*x) + (1/2)*(c^2 - 1/x^2)*(a + b*asec(c*x))^2]
    @test_int [(a + b*asec(c*x))^2/x^4, x, 5, (2*b^2)/(27*x^3) + (4*b^2*c^2)/(9*x) + (4/9)*b*c^3*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x)) + (2*b*c*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x)))/(9*x^2) - (a + b*asec(c*x))^2/(3*x^3)]
    @test_int [(a + b*asec(c*x))^2/x^5, x, 5, b^2/(32*x^4) + (3*b^2*c^2)/(32*x^2) + (3/16)*a*b*c^4*asec(c*x) + (3/32)*b^2*c^4*asec(c*x)^2 + (b*c*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x)))/(8*x^3) + (3*b*c^3*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x)))/(16*x) - (a + b*asec(c*x))^2/(4*x^4)]

    @test_int [x^3*(a + b*asec(c*x))^3, x, 10, -((b^3*sqrt(1 - 1/(c^2*x^2))*x)/(4*c^3)) + (b^2*x^2*(a + b*asec(c*x)))/(4*c^2) + (im*b*(a + b*asec(c*x))^2)/(2*c^4) - (b*sqrt(1 - 1/(c^2*x^2))*x*(a + b*asec(c*x))^2)/(2*c^3) - (b*sqrt(1 - 1/(c^2*x^2))*x^3*(a + b*asec(c*x))^2)/(4*c) + (1/4)*x^4*(a + b*asec(c*x))^3 - (b^2*(a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))))/c^4 + (im*b^3*PolyLog(2, -ℯ^(2*im*asec(c*x))))/(2*c^4)]
    @test_int [x^2*(a + b*asec(c*x))^3, x, 11, (b^2*x*(a + b*asec(c*x)))/c^2 - (b*sqrt(1 - 1/(c^2*x^2))*x^2*(a + b*asec(c*x))^2)/(2*c) + (1/3)*x^3*(a + b*asec(c*x))^3 + (im*b*(a + b*asec(c*x))^2*atan(ℯ^(im*asec(c*x))))/c^3 - (b^3*atanh(sqrt(1 - 1/(c^2*x^2))))/c^3 - (im*b^2*(a + b*asec(c*x))*PolyLog(2, (-im)*ℯ^(im*asec(c*x))))/c^3 + (im*b^2*(a + b*asec(c*x))*PolyLog(2, im*ℯ^(im*asec(c*x))))/c^3 + (b^3*PolyLog(3, (-im)*ℯ^(im*asec(c*x))))/c^3 - (b^3*PolyLog(3, im*ℯ^(im*asec(c*x))))/c^3]
    @test_int [x^1*(a + b*asec(c*x))^3, x, 7, (3*im*b*(a + b*asec(c*x))^2)/(2*c^2) - (3*b*sqrt(1 - 1/(c^2*x^2))*x*(a + b*asec(c*x))^2)/(2*c) + (1/2)*x^2*(a + b*asec(c*x))^3 - (3*b^2*(a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))))/c^2 + (3*im*b^3*PolyLog(2, -ℯ^(2*im*asec(c*x))))/(2*c^2)]
    @test_int [x^0*(a + b*asec(c*x))^3, x, 9, x*(a + b*asec(c*x))^3 + (6*im*b*(a + b*asec(c*x))^2*atan(ℯ^(im*asec(c*x))))/c - (6*im*b^2*(a + b*asec(c*x))*PolyLog(2, (-im)*ℯ^(im*asec(c*x))))/c + (6*im*b^2*(a + b*asec(c*x))*PolyLog(2, im*ℯ^(im*asec(c*x))))/c + (6*b^3*PolyLog(3, (-im)*ℯ^(im*asec(c*x))))/c - (6*b^3*PolyLog(3, im*ℯ^(im*asec(c*x))))/c]
    @test_int [(a + b*asec(c*x))^3/x^1, x, 7, (im*(a + b*asec(c*x))^4)/(4*b) - (a + b*asec(c*x))^3*log(1 + ℯ^(2*im*asec(c*x))) + (3/2)*im*b*(a + b*asec(c*x))^2*PolyLog(2, -ℯ^(2*im*asec(c*x))) - (3/2)*b^2*(a + b*asec(c*x))*PolyLog(3, -ℯ^(2*im*asec(c*x))) - (3/4)*im*b^3*PolyLog(4, -ℯ^(2*im*asec(c*x)))]
    @test_int [(a + b*asec(c*x))^3/x^2, x, 5, -6*b^3*c*sqrt(1 - 1/(c^2*x^2)) + (6*b^2*(a + b*asec(c*x)))/x + 3*b*c*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x))^2 - (a + b*asec(c*x))^3/x]
    @test_int [(a + b*asec(c*x))^3/x^3, x, 6, -((3*b^3*c*sqrt(1 - 1/(c^2*x^2)))/(8*x)) + (3/8)*b^3*c^2*asec(c*x) - (3/4)*b^2*(c^2 - 1/x^2)*(a + b*asec(c*x)) + (3*b*c*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x))^2)/(4*x) - (1/4)*c^2*(a + b*asec(c*x))^3 + (1/2)*(c^2 - 1/x^2)*(a + b*asec(c*x))^3]
    @test_int [(a + b*asec(c*x))^3/x^4, x, 8, (-(14/9))*b^3*c^3*sqrt(1 - 1/(c^2*x^2)) + (2/27)*b^3*c^3*(1 - 1/(c^2*x^2))^(3/2) + (2*b^2*(a + b*asec(c*x)))/(9*x^3) + (4*b^2*c^2*(a + b*asec(c*x)))/(3*x) + (2/3)*b*c^3*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x))^2 + (b*c*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x))^2)/(3*x^2) - (a + b*asec(c*x))^3/(3*x^3)]
    @test_int [(a + b*asec(c*x))^3/x^5, x, 10, -((3*b^3*c*sqrt(1 - 1/(c^2*x^2)))/(128*x^3)) - (45*b^3*c^3*sqrt(1 - 1/(c^2*x^2)))/(256*x) - (45/256)*b^3*c^4*asec(c*x) + (3*b^2*(a + b*asec(c*x)))/(32*x^4) + (9*b^2*c^2*(a + b*asec(c*x)))/(32*x^2) + (3*b*c*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x))^2)/(16*x^3) + (9*b*c^3*sqrt(1 - 1/(c^2*x^2))*(a + b*asec(c*x))^2)/(32*x) + (3/32)*c^4*(a + b*asec(c*x))^3 - (a + b*asec(c*x))^3/(4*x^4)]

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

    @test_int [x^1/(a + b*asec(c*x)), x, 0, Unintegrable(x/(a + b*asec(c*x)), x)]
    @test_int [x^0/(a + b*asec(c*x)), x, 0, Unintegrable(1/(a + b*asec(c*x)), x)]
    @test_int [1/(x^1*(a + b*asec(c*x))), x, 0, Unintegrable(1/(x*(a + b*asec(c*x))), x)]
    @test_int [1/(x^2*(a + b*asec(c*x))), x, 4, -((c*CosIntegral(a/b + asec(c*x))*sin(a/b))/b) + (c*cos(a/b)*SinIntegral(a/b + asec(c*x)))/b]
    @test_int [1/(x^3*(a + b*asec(c*x))), x, 6, -((c^2*CosIntegral((2*a)/b + 2*asec(c*x))*sin((2*a)/b))/(2*b)) + (c^2*cos((2*a)/b)*SinIntegral((2*a)/b + 2*asec(c*x)))/(2*b)]
    @test_int [1/(x^4*(a + b*asec(c*x))), x, 9, -((c^3*CosIntegral(a/b + asec(c*x))*sin(a/b))/(4*b)) - (c^3*CosIntegral((3*a)/b + 3*asec(c*x))*sin((3*a)/b))/(4*b) + (c^3*cos(a/b)*SinIntegral(a/b + asec(c*x)))/(4*b) + (c^3*cos((3*a)/b)*SinIntegral((3*a)/b + 3*asec(c*x)))/(4*b)]

    @test_int [x^1/(a + b*asec(c*x))^2, x, 0, Unintegrable(x/(a + b*asec(c*x))^2, x)]
    @test_int [x^0/(a + b*asec(c*x))^2, x, 0, Unintegrable(1/(a + b*asec(c*x))^2, x)]
    @test_int [1/(x^1*(a + b*asec(c*x))^2), x, 0, Unintegrable(1/(x*(a + b*asec(c*x))^2), x)]
    @test_int [1/(x^2*(a + b*asec(c*x))^2), x, 5, -((c*sqrt(1 - 1/(c^2*x^2)))/(b*(a + b*asec(c*x)))) + (c*cos(a/b)*CosIntegral(a/b + asec(c*x)))/b^2 + (c*sin(a/b)*SinIntegral(a/b + asec(c*x)))/b^2]
    @test_int [1/(x^3*(a + b*asec(c*x))^2), x, 7, (c^2*cos((2*a)/b)*CosIntegral((2*a)/b + 2*asec(c*x)))/b^2 - (c^2*sin(2*asec(c*x)))/(2*b*(a + b*asec(c*x))) + (c^2*sin((2*a)/b)*SinIntegral((2*a)/b + 2*asec(c*x)))/b^2]
    @test_int [1/(x^4*(a + b*asec(c*x))^2), x, 11, -((c^3*sqrt(1 - 1/(c^2*x^2)))/(4*b*(a + b*asec(c*x)))) + (c^3*cos(a/b)*CosIntegral(a/b + asec(c*x)))/(4*b^2) + (3*c^3*cos((3*a)/b)*CosIntegral((3*a)/b + 3*asec(c*x)))/(4*b^2) - (c^3*sin(3*asec(c*x)))/(4*b*(a + b*asec(c*x))) + (c^3*sin(a/b)*SinIntegral(a/b + asec(c*x)))/(4*b^2) + (3*c^3*sin((3*a)/b)*SinIntegral((3*a)/b + 3*asec(c*x)))/(4*b^2)]

    @test_int [x^1/(a + b*asec(c*x))^3, x, 0, Unintegrable(x/(a + b*asec(c*x))^3, x)]
    @test_int [x^0/(a + b*asec(c*x))^3, x, 0, Unintegrable(1/(a + b*asec(c*x))^3, x)]
    @test_int [1/(x^1*(a + b*asec(c*x))^3), x, 0, Unintegrable(1/(x*(a + b*asec(c*x))^3), x)]
    @test_int [1/(x^2*(a + b*asec(c*x))^3), x, 6, -((c*sqrt(1 - 1/(c^2*x^2)))/(2*b*(a + b*asec(c*x))^2)) - 1/(2*b^2*x*(a + b*asec(c*x))) + (c*CosIntegral(a/b + asec(c*x))*sin(a/b))/(2*b^3) - (c*cos(a/b)*SinIntegral(a/b + asec(c*x)))/(2*b^3)]
    @test_int [1/(x^3*(a + b*asec(c*x))^3), x, 8, -((c^2*cos(2*asec(c*x)))/(2*b^2*(a + b*asec(c*x)))) + (c^2*CosIntegral((2*a)/b + 2*asec(c*x))*sin((2*a)/b))/b^3 - (c^2*sin(2*asec(c*x)))/(4*b*(a + b*asec(c*x))^2) - (c^2*cos((2*a)/b)*SinIntegral((2*a)/b + 2*asec(c*x)))/b^3]
    @test_int [1/(x^4*(a + b*asec(c*x))^3), x, 13, -((c^3*sqrt(1 - 1/(c^2*x^2)))/(8*b*(a + b*asec(c*x))^2)) - c^2/(8*b^2*x*(a + b*asec(c*x))) - (3*c^3*cos(3*asec(c*x)))/(8*b^2*(a + b*asec(c*x))) + (c^3*CosIntegral(a/b + asec(c*x))*sin(a/b))/(8*b^3) + (9*c^3*CosIntegral((3*a)/b + 3*asec(c*x))*sin((3*a)/b))/(8*b^3) - (c^3*sin(3*asec(c*x)))/(8*b*(a + b*asec(c*x))^2) - (c^3*cos(a/b)*SinIntegral(a/b + asec(c*x)))/(8*b^3) - (9*c^3*cos((3*a)/b)*SinIntegral((3*a)/b + 3*asec(c*x)))/(8*b^3)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d*x)^m*(a+b*asec(c*x))^n*with*m*symbolic=#

    @test_int [(d*x)^m*(a + b*asec(c*x))^3, x, 0, Unintegrable((d*x)^m*(a + b*asec(c*x))^3, x)]
    @test_int [(d*x)^m*(a + b*asec(c*x))^2, x, 0, Unintegrable((d*x)^m*(a + b*asec(c*x))^2, x)]
    @test_int [(d*x)^m*(a + b*asec(c*x))^1, x, 3, ((d*x)^(1 + m)*(a + b*asec(c*x)))/(d*(1 + m)) - (b*(d*x)^m*HypergeometricFunctions._₂F₁(1/2, -(m/2), 1 - m/2, 1/(c^2*x^2)))/(c*m*(1 + m))]
    @test_int [(d*x)^m/(a + b*asec(c*x))^1, x, 0, Unintegrable((d*x)^m/(a + b*asec(c*x)), x)]
    @test_int [(d*x)^m/(a + b*asec(c*x))^2, x, 0, Unintegrable((d*x)^m/(a + b*asec(c*x))^2, x)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(f*x)^m*(d+e*x)^p*(a+b*asec(c*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(a+b*asec(c*x))^n=#

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

    @test_int [(d + e*x)^3*(a + b*asec(c*x)), x, 11, -((b*e*(9*c^2*d^2 + e^2)*sqrt(1 - 1/(c^2*x^2))*x)/(6*c^3)) - (b*d*e^2*sqrt(1 - 1/(c^2*x^2))*x^2)/(2*c) - (b*e^3*sqrt(1 - 1/(c^2*x^2))*x^3)/(12*c) + (b*d^4*acsc(c*x))/(4*e) + ((d + e*x)^4*(a + b*asec(c*x)))/(4*e) - (b*d*(2*c^2*d^2 + e^2)*atanh(sqrt(1 - 1/(c^2*x^2))))/(2*c^3)]
    @test_int [(d + e*x)^2*(a + b*asec(c*x)), x, 10, -((b*d*e*sqrt(1 - 1/(c^2*x^2))*x)/c) - (b*e^2*sqrt(1 - 1/(c^2*x^2))*x^2)/(6*c) + (b*d^3*acsc(c*x))/(3*e) + ((d + e*x)^3*(a + b*asec(c*x)))/(3*e) - (b*(6*c^2*d^2 + e^2)*atanh(sqrt(1 - 1/(c^2*x^2))))/(6*c^3)]
    @test_int [(d + e*x)^1*(a + b*asec(c*x)), x, 9, -((b*e*sqrt(1 - 1/(c^2*x^2))*x)/(2*c)) + (b*d^2*acsc(c*x))/(2*e) + ((d + e*x)^2*(a + b*asec(c*x)))/(2*e) - (b*d*atanh(sqrt(1 - 1/(c^2*x^2))))/c]
    @test_int [(d + e*x)^0*(a + b*asec(c*x)), x, 5, a*x + b*x*asec(c*x) - (b*atanh(sqrt(1 - 1/(c^2*x^2))))/c]
    @test_int [(a + b*asec(c*x))/(d + e*x)^1, x, 4, ((a + b*asec(c*x))*log(1 + ((e - sqrt((-c^2)*d^2 + e^2))*ℯ^(im*asec(c*x)))/(c*d)))/e + ((a + b*asec(c*x))*log(1 + ((e + sqrt((-c^2)*d^2 + e^2))*ℯ^(im*asec(c*x)))/(c*d)))/e - ((a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))))/e - (im*b*PolyLog(2, -(((e - sqrt((-c^2)*d^2 + e^2))*ℯ^(im*asec(c*x)))/(c*d))))/e - (im*b*PolyLog(2, -(((e + sqrt((-c^2)*d^2 + e^2))*ℯ^(im*asec(c*x)))/(c*d))))/e + (im*b*PolyLog(2, -ℯ^(2*im*asec(c*x))))/(2*e)]
    @test_int [(a + b*asec(c*x))/(d + e*x)^2, x, 7, -((b*acsc(c*x))/(d*e)) - (a + b*asec(c*x))/(e*(d + e*x)) - (b*atanh((c^2*d + e/x)/(c*sqrt(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2)))))/(d*sqrt(c^2*d^2 - e^2))]
    @test_int [(a + b*asec(c*x))/(d + e*x)^3, x, 8, (b*c*e*sqrt(1 - 1/(c^2*x^2)))/(2*d*(c^2*d^2 - e^2)*(e + d/x)) - (b*acsc(c*x))/(2*d^2*e) - (a + b*asec(c*x))/(2*e*(d + e*x)^2) - (b*(2*c^2*d^2 - e^2)*atanh((c^2*d + e/x)/(c*sqrt(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2)))))/(2*d^2*(c^2*d^2 - e^2)^(3/2))]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^(m/2)*(a+b*asec(c*x))^n=#

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

    @test_int [(d + e*x)^(3/2)*(a + b*asec(c*x)), x, 22, (4*b*e*sqrt(d + e*x)*(1 - c^2*x^2))/(15*c^3*sqrt(1 - 1/(c^2*x^2))*x) + (2*(d + e*x)^(5/2)*(a + b*asec(c*x)))/(5*e) + (28*b*d*sqrt(d + e*x)*sqrt(1 - c^2*x^2)*Elliptic.E(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(15*c^2*sqrt(1 - 1/(c^2*x^2))*x*sqrt((c*(d + e*x))/(c*d + e))) + (4*b*(2*c^2*d^2 + e^2)*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.F(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(15*c^4*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) + (4*b*d^3*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.Pi(2, asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(5*c*e*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x))]
    @test_int [(d + e*x)^(1/2)*(a + b*asec(c*x)), x, 15, (2*(d + e*x)^(3/2)*(a + b*asec(c*x)))/(3*e) + (4*b*sqrt(d + e*x)*sqrt(1 - c^2*x^2)*Elliptic.E(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(3*c^2*sqrt(1 - 1/(c^2*x^2))*x*sqrt((c*(d + e*x))/(c*d + e))) + (4*b*d*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.F(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(3*c^2*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) + (4*b*d^2*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.Pi(2, asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(3*c*e*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x))]
    @test_int [(a + b*asec(c*x))/(d + e*x)^(1/2), x, 9, (2*sqrt(d + e*x)*(a + b*asec(c*x)))/e + (4*b*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.F(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(c^2*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) + (4*b*d*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.Pi(2, asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(c*e*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x))]
    @test_int [(a + b*asec(c*x))/(d + e*x)^(3/2), x, 6, -((2*(a + b*asec(c*x)))/(e*sqrt(d + e*x))) - (4*b*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.Pi(2, asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(c*e*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x))]
    @test_int [(a + b*asec(c*x))/(d + e*x)^(5/2), x, 12, -((4*b*e*(1 - c^2*x^2))/(3*c*d*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x))) - (2*(a + b*asec(c*x)))/(3*e*(d + e*x)^(3/2)) + (4*b*sqrt(d + e*x)*sqrt(1 - c^2*x^2)*Elliptic.E(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(3*d*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*sqrt((c*(d + e*x))/(c*d + e))) - (4*b*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.Pi(2, asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(3*c*d*e*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x))]
    @test_int [(a + b*asec(c*x))/(d + e*x)^(7/2), x, 19, -((4*b*e*(1 - c^2*x^2))/(15*c*d*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*(d + e*x)^(3/2))) - (16*b*c*e*(1 - c^2*x^2))/(15*(c^2*d^2 - e^2)^2*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) - (4*b*e*(1 - c^2*x^2))/(5*c*d^2*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) - (2*(a + b*asec(c*x)))/(5*e*(d + e*x)^(5/2)) + (4*b*(7*c^2*d^2 - 3*e^2)*sqrt(d + e*x)*sqrt(1 - c^2*x^2)*Elliptic.E(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(15*(c^2*d^3 - d*e^2)^2*sqrt(1 - 1/(c^2*x^2))*x*sqrt((c*(d + e*x))/(c*d + e))) - (4*b*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.F(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(15*d*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) - (4*b*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.Pi(2, asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(5*c*d^2*e*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)), -((4*b*e*(1 - c^2*x^2))/(15*c*d*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*(d + e*x)^(3/2))) - (16*b*c*e*(1 - c^2*x^2))/(15*(c^2*d^2 - e^2)^2*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) - (4*b*e*(1 - c^2*x^2))/(5*c*d^2*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) - (2*(a + b*asec(c*x)))/(5*e*(d + e*x)^(5/2)) + (16*b*c^2*sqrt(d + e*x)*sqrt(1 - c^2*x^2)*Elliptic.E(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(15*(c^2*d^2 - e^2)^2*sqrt(1 - 1/(c^2*x^2))*x*sqrt((c*(d + e*x))/(c*d + e))) + (4*b*sqrt(d + e*x)*sqrt(1 - c^2*x^2)*Elliptic.E(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(5*d^2*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*sqrt((c*(d + e*x))/(c*d + e))) - (4*b*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.F(asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(15*d*(c^2*d^2 - e^2)*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)) - (4*b*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.Pi(2, asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(5*c*d^2*e*sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x))]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(f*x)^m*(d+e*x^2)^p*(a+b*asec(c*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x^2)^p*(a+b*asec(c*x))=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^4*(d + e*x^2)*(a + b*asec(c*x)), x, 7, -((b*(42*c^2*d + 25*e)*x^2*sqrt(-1 + c^2*x^2))/(560*c^5*sqrt(c^2*x^2))) - (b*(42*c^2*d + 25*e)*x^4*sqrt(-1 + c^2*x^2))/(840*c^3*sqrt(c^2*x^2)) - (b*e*x^6*sqrt(-1 + c^2*x^2))/(42*c*sqrt(c^2*x^2)) + (1/5)*d*x^5*(a + b*asec(c*x)) + (1/7)*e*x^7*(a + b*asec(c*x)) - (b*(42*c^2*d + 25*e)*x*atanh((c*x)/sqrt(-1 + c^2*x^2)))/(560*c^6*sqrt(c^2*x^2))]
    @test_int [x^2*(d + e*x^2)*(a + b*asec(c*x)), x, 6, -((b*(20*c^2*d + 9*e)*x^2*sqrt(-1 + c^2*x^2))/(120*c^3*sqrt(c^2*x^2))) - (b*e*x^4*sqrt(-1 + c^2*x^2))/(20*c*sqrt(c^2*x^2)) + (1/3)*d*x^3*(a + b*asec(c*x)) + (1/5)*e*x^5*(a + b*asec(c*x)) - (b*(20*c^2*d + 9*e)*x*atanh((c*x)/sqrt(-1 + c^2*x^2)))/(120*c^4*sqrt(c^2*x^2))]
    @test_int [x^0*(d + e*x^2)*(a + b*asec(c*x)), x, 5, -((b*e*x^2*sqrt(-1 + c^2*x^2))/(6*c*sqrt(c^2*x^2))) + d*x*(a + b*asec(c*x)) + (1/3)*e*x^3*(a + b*asec(c*x)) - (b*(6*c^2*d + e)*x*atanh((c*x)/sqrt(-1 + c^2*x^2)))/(6*c^2*sqrt(c^2*x^2))]
    @test_int [(d + e*x^2)*(a + b*asec(c*x))/x^2, x, 4, (b*c*d*sqrt(-1 + c^2*x^2))/sqrt(c^2*x^2) - (d*(a + b*asec(c*x)))/x + e*x*(a + b*asec(c*x)) - (b*e*x*atanh((c*x)/sqrt(-1 + c^2*x^2)))/sqrt(c^2*x^2)]
    @test_int [(d + e*x^2)*(a + b*asec(c*x))/x^4, x, 4, (b*c*(2*c^2*d + 9*e)*sqrt(-1 + c^2*x^2))/(9*sqrt(c^2*x^2)) + (b*c*d*sqrt(-1 + c^2*x^2))/(9*x^2*sqrt(c^2*x^2)) - (d*(a + b*asec(c*x)))/(3*x^3) - (e*(a + b*asec(c*x)))/x]
    @test_int [(d + e*x^2)*(a + b*asec(c*x))/x^6, x, 5, (2*b*c^3*(12*c^2*d + 25*e)*sqrt(-1 + c^2*x^2))/(225*sqrt(c^2*x^2)) + (b*c*d*sqrt(-1 + c^2*x^2))/(25*x^4*sqrt(c^2*x^2)) + (b*c*(12*c^2*d + 25*e)*sqrt(-1 + c^2*x^2))/(225*x^2*sqrt(c^2*x^2)) - (d*(a + b*asec(c*x)))/(5*x^5) - (e*(a + b*asec(c*x)))/(3*x^3)]
    @test_int [(d + e*x^2)*(a + b*asec(c*x))/x^8, x, 6, (8*b*c^5*(30*c^2*d + 49*e)*sqrt(-1 + c^2*x^2))/(3675*sqrt(c^2*x^2)) + (b*c*d*sqrt(-1 + c^2*x^2))/(49*x^6*sqrt(c^2*x^2)) + (b*c*(30*c^2*d + 49*e)*sqrt(-1 + c^2*x^2))/(1225*x^4*sqrt(c^2*x^2)) + (4*b*c^3*(30*c^2*d + 49*e)*sqrt(-1 + c^2*x^2))/(3675*x^2*sqrt(c^2*x^2)) - (d*(a + b*asec(c*x)))/(7*x^7) - (e*(a + b*asec(c*x)))/(5*x^5)]

    @test_int [x^5*(d + e*x^2)*(a + b*asec(c*x)), x, 5, -((b*(4*c^2*d + 3*e)*x*sqrt(-1 + c^2*x^2))/(24*c^7*sqrt(c^2*x^2))) - (b*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*sqrt(c^2*x^2)) - (b*(4*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*sqrt(c^2*x^2)) - (b*e*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*sqrt(c^2*x^2)) + (1/6)*d*x^6*(a + b*asec(c*x)) + (1/8)*e*x^8*(a + b*asec(c*x))]
    @test_int [x^3*(d + e*x^2)*(a + b*asec(c*x)), x, 5, -((b*(3*c^2*d + 2*e)*x*sqrt(-1 + c^2*x^2))/(12*c^5*sqrt(c^2*x^2))) - (b*(3*c^2*d + 4*e)*x*(-1 + c^2*x^2)^(3/2))/(36*c^5*sqrt(c^2*x^2)) - (b*e*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*sqrt(c^2*x^2)) + (1/4)*d*x^4*(a + b*asec(c*x)) + (1/6)*e*x^6*(a + b*asec(c*x))]
    @test_int [x^1*(d + e*x^2)*(a + b*asec(c*x)), x, 6, -((b*(2*c^2*d + e)*x*sqrt(-1 + c^2*x^2))/(4*c^3*sqrt(c^2*x^2))) - (b*e*x*(-1 + c^2*x^2)^(3/2))/(12*c^3*sqrt(c^2*x^2)) + ((d + e*x^2)^2*(a + b*asec(c*x)))/(4*e) - (b*c*d^2*x*atan(sqrt(-1 + c^2*x^2)))/(4*e*sqrt(c^2*x^2))]
    @test_int [(d + e*x^2)*(a + b*asec(c*x))/x^1, x, 11, -((b*e*sqrt(1 - 1/(c^2*x^2))*x)/(2*c)) - (1/2)*im*b*d*acsc(c*x)^2 + (1/2)*e*x^2*(a + b*asec(c*x)) + b*d*acsc(c*x)*log(1 - ℯ^(2*im*acsc(c*x))) - b*d*acsc(c*x)*log(1/x) - d*(a + b*asec(c*x))*log(1/x) - (1/2)*im*b*d*PolyLog(2, ℯ^(2*im*acsc(c*x)))]
    @test_int [(d + e*x^2)*(a + b*asec(c*x))/x^3, x, 13, (b*c*d*sqrt(1 - 1/(c^2*x^2)))/(4*x) - (1/4)*b*c^2*d*acsc(c*x) - (1/2)*im*b*e*acsc(c*x)^2 - (d*(a + b*asec(c*x)))/(2*x^2) + b*e*acsc(c*x)*log(1 - ℯ^(2*im*acsc(c*x))) - b*e*acsc(c*x)*log(1/x) - e*(a + b*asec(c*x))*log(1/x) - (1/2)*im*b*e*PolyLog(2, ℯ^(2*im*acsc(c*x)))]

    @test_int [x^2*(d + e*x^2)^2*(a + b*asec(c*x)), x, 7, -((b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x^2*sqrt(-1 + c^2*x^2))/(1680*c^5*sqrt(c^2*x^2))) - (b*e*(84*c^2*d + 25*e)*x^4*sqrt(-1 + c^2*x^2))/(840*c^3*sqrt(c^2*x^2)) - (b*e^2*x^6*sqrt(-1 + c^2*x^2))/(42*c*sqrt(c^2*x^2)) + (1/3)*d^2*x^3*(a + b*asec(c*x)) + (2/5)*d*e*x^5*(a + b*asec(c*x)) + (1/7)*e^2*x^7*(a + b*asec(c*x)) - (b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x*atanh((c*x)/sqrt(-1 + c^2*x^2)))/(1680*c^6*sqrt(c^2*x^2))]
    @test_int [x^0*(d + e*x^2)^2*(a + b*asec(c*x)), x, 6, -((b*e*(40*c^2*d + 9*e)*x^2*sqrt(-1 + c^2*x^2))/(120*c^3*sqrt(c^2*x^2))) - (b*e^2*x^4*sqrt(-1 + c^2*x^2))/(20*c*sqrt(c^2*x^2)) + d^2*x*(a + b*asec(c*x)) + (2/3)*d*e*x^3*(a + b*asec(c*x)) + (1/5)*e^2*x^5*(a + b*asec(c*x)) - (b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*x*atanh((c*x)/sqrt(-1 + c^2*x^2)))/(120*c^4*sqrt(c^2*x^2))]
    @test_int [(d + e*x^2)^2*(a + b*asec(c*x))/x^2, x, 6, (b*c*d^2*sqrt(-1 + c^2*x^2))/sqrt(c^2*x^2) - (b*e^2*x^2*sqrt(-1 + c^2*x^2))/(6*c*sqrt(c^2*x^2)) - (d^2*(a + b*asec(c*x)))/x + 2*d*e*x*(a + b*asec(c*x)) + (1/3)*e^2*x^3*(a + b*asec(c*x)) - (b*e*(12*c^2*d + e)*x*atanh((c*x)/sqrt(-1 + c^2*x^2)))/(6*c^2*sqrt(c^2*x^2))]
    @test_int [(d + e*x^2)^2*(a + b*asec(c*x))/x^4, x, 6, (2*b*c*d*(c^2*d + 9*e)*sqrt(-1 + c^2*x^2))/(9*sqrt(c^2*x^2)) + (b*c*d^2*sqrt(-1 + c^2*x^2))/(9*x^2*sqrt(c^2*x^2)) - (d^2*(a + b*asec(c*x)))/(3*x^3) - (2*d*e*(a + b*asec(c*x)))/x + e^2*x*(a + b*asec(c*x)) - (b*e^2*x*atanh((c*x)/sqrt(-1 + c^2*x^2)))/sqrt(c^2*x^2)]
    @test_int [(d + e*x^2)^2*(a + b*asec(c*x))/x^6, x, 5, (b*c*(24*c^4*d^2 + 100*c^2*d*e + 225*e^2)*sqrt(-1 + c^2*x^2))/(225*sqrt(c^2*x^2)) + (b*c*d^2*sqrt(-1 + c^2*x^2))/(25*x^4*sqrt(c^2*x^2)) + (2*b*c*d*(6*c^2*d + 25*e)*sqrt(-1 + c^2*x^2))/(225*x^2*sqrt(c^2*x^2)) - (d^2*(a + b*asec(c*x)))/(5*x^5) - (2*d*e*(a + b*asec(c*x)))/(3*x^3) - (e^2*(a + b*asec(c*x)))/x]
    @test_int [(d + e*x^2)^2*(a + b*asec(c*x))/x^8, x, 6, (2*b*c^3*(360*c^4*d^2 + 1176*c^2*d*e + 1225*e^2)*sqrt(-1 + c^2*x^2))/(11025*sqrt(c^2*x^2)) + (b*c*d^2*sqrt(-1 + c^2*x^2))/(49*x^6*sqrt(c^2*x^2)) + (2*b*c*d*(15*c^2*d + 49*e)*sqrt(-1 + c^2*x^2))/(1225*x^4*sqrt(c^2*x^2)) + (b*c*(360*c^4*d^2 + 1176*c^2*d*e + 1225*e^2)*sqrt(-1 + c^2*x^2))/(11025*x^2*sqrt(c^2*x^2)) - (d^2*(a + b*asec(c*x)))/(7*x^7) - (2*d*e*(a + b*asec(c*x)))/(5*x^5) - (e^2*(a + b*asec(c*x)))/(3*x^3)]

    @test_int [x^3*(d + e*x^2)^2*(a + b*asec(c*x)), x, 5, -((b*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*x*sqrt(-1 + c^2*x^2))/(24*c^7*sqrt(c^2*x^2))) - (b*(6*c^4*d^2 + 16*c^2*d*e + 9*e^2)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*sqrt(c^2*x^2)) - (b*e*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*sqrt(c^2*x^2)) - (b*e^2*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*sqrt(c^2*x^2)) + (1/4)*d^2*x^4*(a + b*asec(c*x)) + (1/3)*d*e*x^6*(a + b*asec(c*x)) + (1/8)*e^2*x^8*(a + b*asec(c*x))]
    @test_int [x^1*(d + e*x^2)^2*(a + b*asec(c*x)), x, 6, -((b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*x*sqrt(-1 + c^2*x^2))/(6*c^5*sqrt(c^2*x^2))) - (b*e*(3*c^2*d + 2*e)*x*(-1 + c^2*x^2)^(3/2))/(18*c^5*sqrt(c^2*x^2)) - (b*e^2*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*sqrt(c^2*x^2)) + ((d + e*x^2)^3*(a + b*asec(c*x)))/(6*e) - (b*c*d^3*x*atan(sqrt(-1 + c^2*x^2)))/(6*e*sqrt(c^2*x^2))]
    @test_int [(d + e*x^2)^2*(a + b*asec(c*x))/x^1, x, 12, -((b*e*(6*c^2*d + e)*sqrt(1 - 1/(c^2*x^2))*x)/(6*c^3)) - (b*e^2*sqrt(1 - 1/(c^2*x^2))*x^3)/(12*c) - (1/2)*im*b*d^2*acsc(c*x)^2 + d*e*x^2*(a + b*asec(c*x)) + (1/4)*e^2*x^4*(a + b*asec(c*x)) + b*d^2*acsc(c*x)*log(1 - ℯ^(2*im*acsc(c*x))) - b*d^2*acsc(c*x)*log(1/x) - d^2*(a + b*asec(c*x))*log(1/x) - (1/2)*im*b*d^2*PolyLog(2, ℯ^(2*im*acsc(c*x)))]
    @test_int [(d + e*x^2)^2*(a + b*asec(c*x))/x^3, x, 14, (b*c*d^2*sqrt(1 - 1/(c^2*x^2)))/(4*x) - (b*e^2*sqrt(1 - 1/(c^2*x^2))*x)/(2*c) - (1/4)*b*c^2*d^2*acsc(c*x) - im*b*d*e*acsc(c*x)^2 - (d^2*(a + b*asec(c*x)))/(2*x^2) + (1/2)*e^2*x^2*(a + b*asec(c*x)) + 2*b*d*e*acsc(c*x)*log(1 - ℯ^(2*im*acsc(c*x))) - 2*b*d*e*acsc(c*x)*log(1/x) - 2*d*e*(a + b*asec(c*x))*log(1/x) - im*b*d*e*PolyLog(2, ℯ^(2*im*acsc(c*x)))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    #= [x^3*(a + b*asec(c*x))/(d + e*x^2), x, 47, -((b*sqrt(1 - 1/(c^2*x^2))*x)/(2*c*e)) + (im*b*d*asec(c*x)^2)/(2*e^2) + (x^2*(a + b*asec(c*x)))/(2*e) - (im*d*(a + b*asec(c*x))^2)/(2*b*e^2) - (b*d*asec(c*x)*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^2) - (b*d*asec(c*x)*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^2) - (b*d*asec(c*x)*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^2) - (b*d*asec(c*x)*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^2) + (d*(a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))))/e^2 - (a*d*log(sqrt(e) - sqrt(-d)/x))/(2*e^2) - (a*d*log(sqrt(e) + sqrt(-d)/x))/(2*e^2) + (im*b*d*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*e^2) + (im*b*d*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^2) + (im*b*d*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*e^2) + (im*b*d*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^2) - (im*b*d*PolyLog(2, -ℯ^(2*im*asec(c*x))))/(2*e^2)] =#
    @test_int [x^2*(a + b*asec(c*x))/(d + e*x^2), x, 25, (x*(a + b*asec(c*x)))/e - (b*atanh(sqrt(1 - 1/(c^2*x^2))))/(c*e) + (sqrt(-d)*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^(3/2)) - (sqrt(-d)*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^(3/2)) + (sqrt(-d)*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^(3/2)) - (sqrt(-d)*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^(3/2)) + (im*b*sqrt(-d)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*e^(3/2)) - (im*b*sqrt(-d)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^(3/2)) + (im*b*sqrt(-d)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*e^(3/2)) - (im*b*sqrt(-d)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^(3/2))]
    @test_int [x^1*(a + b*asec(c*x))/(d + e*x^2), x, 26, ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e) + ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e) - ((a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))))/e - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*e) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e) - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*e) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e) + (im*b*PolyLog(2, -ℯ^(2*im*asec(c*x))))/(2*e)]
    @test_int [x^0*(a + b*asec(c*x))/(d + e*x^2), x, 19, ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*sqrt(-d)*sqrt(e)) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*sqrt(-d)*sqrt(e)) + ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*sqrt(-d)*sqrt(e)) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*sqrt(-d)*sqrt(e)) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*sqrt(-d)*sqrt(e)) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*sqrt(-d)*sqrt(e)) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*sqrt(-d)*sqrt(e)) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*sqrt(-d)*sqrt(e))]
    @test_int [(a + b*asec(c*x))/(x^1*(d + e*x^2)), x, 19, (im*(a + b*asec(c*x))^2)/(2*b*d) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*d) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*d) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d)]
    @test_int [(a + b*asec(c*x))/(x^2*(d + e*x^2)), x, 24, (b*c*sqrt(1 - 1/(c^2*x^2)))/d - a/(d*x) - (b*asec(c*x))/(d*x) + (sqrt(e)*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*(-d)^(3/2)) - (sqrt(e)*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*(-d)^(3/2)) + (sqrt(e)*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*(-d)^(3/2)) - (sqrt(e)*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*(-d)^(3/2)) + (im*b*sqrt(e)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*(-d)^(3/2)) - (im*b*sqrt(e)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*(-d)^(3/2)) + (im*b*sqrt(e)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*(-d)^(3/2)) - (im*b*sqrt(e)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*(-d)^(3/2))]

    @test_int [x^5*(a + b*asec(c*x))/(d + e*x^2)^2, x, 31, -((b*sqrt(1 - 1/(c^2*x^2))*x)/(2*c*e^2)) + (d*(a + b*asec(c*x)))/(2*e^2*(e + d/x^2)) + (x^2*(a + b*asec(c*x)))/(2*e^2) + (b*d*atan(sqrt(c^2*d + e)/(c*sqrt(e)*sqrt(1 - 1/(c^2*x^2))*x)))/(2*e^(5/2)*sqrt(c^2*d + e)) - (d*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/e^3 - (d*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/e^3 - (d*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/e^3 - (d*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/e^3 + (2*d*(a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))))/e^3 + (im*b*d*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/e^3 + (im*b*d*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/e^3 + (im*b*d*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/e^3 + (im*b*d*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/e^3 - (im*b*d*PolyLog(2, -ℯ^(2*im*asec(c*x))))/e^3]
    @test_int [x^3*(a + b*asec(c*x))/(d + e*x^2)^2, x, 29, -((a + b*asec(c*x))/(2*e*(e + d/x^2))) - (b*atan(sqrt(c^2*d + e)/(c*sqrt(e)*sqrt(1 - 1/(c^2*x^2))*x)))/(2*e^(3/2)*sqrt(c^2*d + e)) + ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^2) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^2) + ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^2) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^2) - ((a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))))/e^2 - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*e^2) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^2) - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*e^2) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^2) + (im*b*PolyLog(2, -ℯ^(2*im*asec(c*x))))/(2*e^2)]
    @test_int [x^1*(a + b*asec(c*x))/(d + e*x^2)^2, x, 7, -((a + b*asec(c*x))/(2*e*(d + e*x^2))) + (b*c*x*atan(sqrt(-1 + c^2*x^2)))/(2*d*e*sqrt(c^2*x^2)) - (b*c*x*atan((sqrt(e)*sqrt(-1 + c^2*x^2))/sqrt(c^2*d + e)))/(2*d*sqrt(e)*sqrt(c^2*d + e)*sqrt(c^2*x^2))]
    @test_int [(a + b*asec(c*x))/(x^1*(d + e*x^2)^2), x, 24, -((e*(a + b*asec(c*x)))/(2*d^2*(e + d/x^2))) + (im*(a + b*asec(c*x))^2)/(2*b*d^2) - (b*sqrt(e)*atan(sqrt(c^2*d + e)/(c*sqrt(e)*sqrt(1 - 1/(c^2*x^2))*x)))/(2*d^2*sqrt(c^2*d + e)) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d^2) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d^2) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d^2) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d^2) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*d^2) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d^2) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*d^2) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d^2)]

    @test_int [x^4*(a + b*asec(c*x))/(d + e*x^2)^2, x, 51, -((d*(a + b*asec(c*x)))/(4*e^2*(sqrt(-d)*sqrt(e) - d/x))) + (d*(a + b*asec(c*x)))/(4*e^2*(sqrt(-d)*sqrt(e) + d/x)) + (x*(a + b*asec(c*x)))/e^2 - (b*atanh(sqrt(1 - 1/(c^2*x^2))))/(c*e^2) - (b*sqrt(d)*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(4*e^2*sqrt(c^2*d + e)) - (b*sqrt(d)*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(4*e^2*sqrt(c^2*d + e)) + (3*sqrt(-d)*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*e^(5/2)) - (3*sqrt(-d)*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*e^(5/2)) + (3*sqrt(-d)*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*e^(5/2)) - (3*sqrt(-d)*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*e^(5/2)) + (3*im*b*sqrt(-d)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(4*e^(5/2)) - (3*im*b*sqrt(-d)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*e^(5/2)) + (3*im*b*sqrt(-d)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(4*e^(5/2)) - (3*im*b*sqrt(-d)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*e^(5/2))]
    @test_int [x^2*(a + b*asec(c*x))/(d + e*x^2)^2, x, 27, (a + b*asec(c*x))/(4*e*(sqrt(-d)*sqrt(e) - d/x)) - (a + b*asec(c*x))/(4*e*(sqrt(-d)*sqrt(e) + d/x)) + (b*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(4*sqrt(d)*e*sqrt(c^2*d + e)) + (b*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(4*sqrt(d)*e*sqrt(c^2*d + e)) + ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*sqrt(-d)*e^(3/2)) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*sqrt(-d)*e^(3/2)) + ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*sqrt(-d)*e^(3/2)) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*sqrt(-d)*e^(3/2)) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(4*sqrt(-d)*e^(3/2)) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*sqrt(-d)*e^(3/2)) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(4*sqrt(-d)*e^(3/2)) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*sqrt(-d)*e^(3/2))]
    @test_int [x^0*(a + b*asec(c*x))/(d + e*x^2)^2, x, 47, -((a + b*asec(c*x))/(4*d*(sqrt(-d)*sqrt(e) - d/x))) + (a + b*asec(c*x))/(4*d*(sqrt(-d)*sqrt(e) + d/x)) - (b*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(4*d^(3/2)*sqrt(c^2*d + e)) - (b*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(4*d^(3/2)*sqrt(c^2*d + e)) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*(-d)^(3/2)*sqrt(e)) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*(-d)^(3/2)*sqrt(e)) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*(-d)^(3/2)*sqrt(e)) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*(-d)^(3/2)*sqrt(e)) - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(4*(-d)^(3/2)*sqrt(e)) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*(-d)^(3/2)*sqrt(e)) - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(4*(-d)^(3/2)*sqrt(e)) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*(-d)^(3/2)*sqrt(e))]
    @test_int [(a + b*asec(c*x))/(x^2*(d + e*x^2)^2), x, 50, (b*c*sqrt(1 - 1/(c^2*x^2)))/d^2 - a/(d^2*x) - (b*asec(c*x))/(d^2*x) + (e*(a + b*asec(c*x)))/(4*d^2*(sqrt(-d)*sqrt(e) - d/x)) - (e*(a + b*asec(c*x)))/(4*d^2*(sqrt(-d)*sqrt(e) + d/x)) + (b*e*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(4*d^(5/2)*sqrt(c^2*d + e)) + (b*e*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(4*d^(5/2)*sqrt(c^2*d + e)) - (3*sqrt(e)*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*(-d)^(5/2)) + (3*sqrt(e)*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*(-d)^(5/2)) - (3*sqrt(e)*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*(-d)^(5/2)) + (3*sqrt(e)*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*(-d)^(5/2)) - (3*im*b*sqrt(e)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(4*(-d)^(5/2)) + (3*im*b*sqrt(e)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(4*(-d)^(5/2)) - (3*im*b*sqrt(e)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(4*(-d)^(5/2)) + (3*im*b*sqrt(e)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(4*(-d)^(5/2))]

    @test_int [x^5*(a + b*asec(c*x))/(d + e*x^2)^3, x, 33, -((b*c*d*sqrt(1 - 1/(c^2*x^2)))/(8*e^2*(c^2*d + e)*(e + d/x^2)*x)) - (a + b*asec(c*x))/(4*e*(e + d/x^2)^2) - (a + b*asec(c*x))/(2*e^2*(e + d/x^2)) - (b*atan(sqrt(c^2*d + e)/(c*sqrt(e)*sqrt(1 - 1/(c^2*x^2))*x)))/(2*e^(5/2)*sqrt(c^2*d + e)) - (b*(c^2*d + 2*e)*atan(sqrt(c^2*d + e)/(c*sqrt(e)*sqrt(1 - 1/(c^2*x^2))*x)))/(8*e^(5/2)*(c^2*d + e)^(3/2)) + ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^3) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^3) + ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^3) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^3) - ((a + b*asec(c*x))*log(1 + ℯ^(2*im*asec(c*x))))/e^3 - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*e^3) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*e^3) - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*e^3) - (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*e^3) + (im*b*PolyLog(2, -ℯ^(2*im*asec(c*x))))/(2*e^3)]
    @test_int [x^3*(a + b*asec(c*x))/(d + e*x^2)^3, x, 6, (b*c*x*sqrt(-1 + c^2*x^2))/(8*e*(c^2*d + e)*sqrt(c^2*x^2)*(d + e*x^2)) + (x^4*(a + b*asec(c*x)))/(4*d*(d + e*x^2)^2) - (b*c*(c^2*d + 2*e)*x*atan((sqrt(e)*sqrt(-1 + c^2*x^2))/sqrt(c^2*d + e)))/(8*d*e^(3/2)*(c^2*d + e)^(3/2)*sqrt(c^2*x^2))]
    @test_int [x^1*(a + b*asec(c*x))/(d + e*x^2)^3, x, 8, -((b*c*x*sqrt(-1 + c^2*x^2))/(8*d*(c^2*d + e)*sqrt(c^2*x^2)*(d + e*x^2))) - (a + b*asec(c*x))/(4*e*(d + e*x^2)^2) + (b*c*x*atan(sqrt(-1 + c^2*x^2)))/(4*d^2*e*sqrt(c^2*x^2)) - (b*c*(3*c^2*d + 2*e)*x*atan((sqrt(e)*sqrt(-1 + c^2*x^2))/sqrt(c^2*d + e)))/(8*d^2*sqrt(e)*(c^2*d + e)^(3/2)*sqrt(c^2*x^2))]
    @test_int [(a + b*asec(c*x))/(x^1*(d + e*x^2)^3), x, 28, (b*c*e*sqrt(1 - 1/(c^2*x^2)))/(8*d^2*(c^2*d + e)*(e + d/x^2)*x) + (e^2*(a + b*asec(c*x)))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*asec(c*x)))/(d^3*(e + d/x^2)) + (im*(a + b*asec(c*x))^2)/(2*b*d^3) - (b*sqrt(e)*atan(sqrt(c^2*d + e)/(c*sqrt(e)*sqrt(1 - 1/(c^2*x^2))*x)))/(d^3*sqrt(c^2*d + e)) + (b*sqrt(e)*(c^2*d + 2*e)*atan(sqrt(c^2*d + e)/(c*sqrt(e)*sqrt(1 - 1/(c^2*x^2))*x)))/(8*d^3*(c^2*d + e)^(3/2)) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d^3) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d^3) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d^3) - ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d^3) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(2*d^3) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(2*d^3) + (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(2*d^3) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(2*d^3)]

    @test_int [x^4*(a + b*asec(c*x))/(d + e*x^2)^3, x, 35, (b*c*sqrt(-d)*sqrt(1 - 1/(c^2*x^2)))/(16*e^(3/2)*(c^2*d + e)*(sqrt(-d)*sqrt(e) - d/x)) + (b*c*sqrt(-d)*sqrt(1 - 1/(c^2*x^2)))/(16*e^(3/2)*(c^2*d + e)*(sqrt(-d)*sqrt(e) + d/x)) + (sqrt(-d)*(a + b*asec(c*x)))/(16*e^(3/2)*(sqrt(-d)*sqrt(e) - d/x)^2) + (3*(a + b*asec(c*x)))/(16*e^2*(sqrt(-d)*sqrt(e) - d/x)) - (sqrt(-d)*(a + b*asec(c*x)))/(16*e^(3/2)*(sqrt(-d)*sqrt(e) + d/x)^2) - (3*(a + b*asec(c*x)))/(16*e^2*(sqrt(-d)*sqrt(e) + d/x)) + (b*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*sqrt(d)*e*(c^2*d + e)^(3/2)) + (3*b*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*sqrt(d)*e^2*sqrt(c^2*d + e)) + (b*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*sqrt(d)*e*(c^2*d + e)^(3/2)) + (3*b*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*sqrt(d)*e^2*sqrt(c^2*d + e)) + (3*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*sqrt(-d)*e^(5/2)) - (3*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*sqrt(-d)*e^(5/2)) + (3*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*sqrt(-d)*e^(5/2)) - (3*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*sqrt(-d)*e^(5/2)) + (3*im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(16*sqrt(-d)*e^(5/2)) - (3*im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*sqrt(-d)*e^(5/2)) + (3*im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(16*sqrt(-d)*e^(5/2)) - (3*im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*sqrt(-d)*e^(5/2))]
    @test_int [x^2*(a + b*asec(c*x))/(d + e*x^2)^3, x, 63, (b*c*sqrt(1 - 1/(c^2*x^2)))/(16*sqrt(-d)*sqrt(e)*(c^2*d + e)*(sqrt(-d)*sqrt(e) - d/x)) + (b*c*sqrt(1 - 1/(c^2*x^2)))/(16*sqrt(-d)*sqrt(e)*(c^2*d + e)*(sqrt(-d)*sqrt(e) + d/x)) + (a + b*asec(c*x))/(16*sqrt(-d)*sqrt(e)*(sqrt(-d)*sqrt(e) - d/x)^2) + (a + b*asec(c*x))/(16*d*e*(sqrt(-d)*sqrt(e) - d/x)) - (a + b*asec(c*x))/(16*sqrt(-d)*sqrt(e)*(sqrt(-d)*sqrt(e) + d/x)^2) - (a + b*asec(c*x))/(16*d*e*(sqrt(-d)*sqrt(e) + d/x)) - (b*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(3/2)*(c^2*d + e)^(3/2)) + (b*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(3/2)*e*sqrt(c^2*d + e)) - (b*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(3/2)*(c^2*d + e)^(3/2)) + (b*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(3/2)*e*sqrt(c^2*d + e)) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*(-d)^(3/2)*e^(3/2)) - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(16*(-d)^(3/2)*e^(3/2)) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*(-d)^(3/2)*e^(3/2)) - (im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(16*(-d)^(3/2)*e^(3/2)) + (im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*(-d)^(3/2)*e^(3/2))]
    @test_int [x^0*(a + b*asec(c*x))/(d + e*x^2)^3, x, 81, (b*c*sqrt(e)*sqrt(1 - 1/(c^2*x^2)))/(16*(-d)^(3/2)*(c^2*d + e)*(sqrt(-d)*sqrt(e) - d/x)) + (b*c*sqrt(e)*sqrt(1 - 1/(c^2*x^2)))/(16*(-d)^(3/2)*(c^2*d + e)*(sqrt(-d)*sqrt(e) + d/x)) + (sqrt(e)*(a + b*asec(c*x)))/(16*(-d)^(3/2)*(sqrt(-d)*sqrt(e) - d/x)^2) - (5*(a + b*asec(c*x)))/(16*d^2*(sqrt(-d)*sqrt(e) - d/x)) - (sqrt(e)*(a + b*asec(c*x)))/(16*(-d)^(3/2)*(sqrt(-d)*sqrt(e) + d/x)^2) + (5*(a + b*asec(c*x)))/(16*d^2*(sqrt(-d)*sqrt(e) + d/x)) + (b*e*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(5/2)*(c^2*d + e)^(3/2)) - (5*b*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(5/2)*sqrt(c^2*d + e)) + (b*e*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(5/2)*(c^2*d + e)^(3/2)) - (5*b*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(5/2)*sqrt(c^2*d + e)) + (3*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*(-d)^(5/2)*sqrt(e)) - (3*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*(-d)^(5/2)*sqrt(e)) + (3*(a + b*asec(c*x))*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*(-d)^(5/2)*sqrt(e)) - (3*(a + b*asec(c*x))*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*(-d)^(5/2)*sqrt(e)) + (3*im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e)))))/(16*(-d)^(5/2)*sqrt(e)) - (3*im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - sqrt(c^2*d + e))))/(16*(-d)^(5/2)*sqrt(e)) + (3*im*b*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e)))))/(16*(-d)^(5/2)*sqrt(e)) - (3*im*b*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + sqrt(c^2*d + e))))/(16*(-d)^(5/2)*sqrt(e))]
    #= [(a + b*asec(c*x))/(x^2*(d + e*x^2)^3), x, 96, (b*c*sqrt(1 - 1/(c^2*x^2)))/d^3 + (b*c*e^(3/2)*sqrt(1 - 1/(c^2*x^2)))/(16*(-d)^(5/2)*(c^2*d + e)*(sqrt(-d)*sqrt(e) - d/x)) + (b*c*e^(3/2)*sqrt(1 - 1/(c^2*x^2)))/(16*(-d)^(5/2)*(c^2*d + e)*(sqrt(-d)*sqrt(e) + d/x)) - a/(d^3*x) + (a*e^2)/(4*d^3*(e + d/x^2)^2*x) - (9*a*e)/(8*d^3*(e + d/x^2)*x) + (15*a*sqrt(e)*acot((sqrt(e)*x)/sqrt(d)))/(8*d^(7/2)) + (b*e^(3/2)*asec(c*x))/(16*(-d)^(5/2)*(sqrt(-d)*sqrt(e) - d/x)^2) + (9*b*e*asec(c*x))/(16*d^3*(sqrt(-d)*sqrt(e) - d/x)) - (b*e^(3/2)*asec(c*x))/(16*(-d)^(5/2)*(sqrt(-d)*sqrt(e) + d/x)^2) - (9*b*e*asec(c*x))/(16*d^3*(sqrt(-d)*sqrt(e) + d/x)) - (b*asec(c*x))/(d^3*x) - (b*e^2*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(7/2)*(c^2*d + e)^(3/2)) + (9*b*e*atanh((c^2*d - (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(7/2)*sqrt(c^2*d + e)) - (b*e^2*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(7/2)*(c^2*d + e)^(3/2)) + (9*b*e*atanh((c^2*d + (sqrt(-d)*sqrt(e))/x)/(c*sqrt(d)*sqrt(c^2*d + e)*sqrt(1 - 1/(c^2*x^2)))))/(16*d^(7/2)*sqrt(c^2*d + e)) + (15*b*sqrt(e)*asec(c*x)*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - c*sqrt(d + e/c^2))))/(16*(-d)^(7/2)) - (15*b*sqrt(e)*asec(c*x)*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - c*sqrt(d + e/c^2))))/(16*(-d)^(7/2)) + (15*b*sqrt(e)*asec(c*x)*log(1 - (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + c*sqrt(d + e/c^2))))/(16*(-d)^(7/2)) - (15*b*sqrt(e)*asec(c*x)*log(1 + (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + c*sqrt(d + e/c^2))))/(16*(-d)^(7/2)) + (15*im*b*sqrt(e)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - c*sqrt(d + e/c^2)))))/(16*(-d)^(7/2)) - (15*im*b*sqrt(e)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) - c*sqrt(d + e/c^2))))/(16*(-d)^(7/2)) + (15*im*b*sqrt(e)*PolyLog(2, -((c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + c*sqrt(d + e/c^2)))))/(16*(-d)^(7/2)) - (15*im*b*sqrt(e)*PolyLog(2, (c*sqrt(-d)*ℯ^(im*asec(c*x)))/(sqrt(e) + c*sqrt(d + e/c^2))))/(16*(-d)^(7/2))] =#

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

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*sqrt(d + e*x^2)*(a + b*asec(c*x)), x, If(13>=8, 12, 13), (b*(23*c^4*d^2 + 12*c^2*d*e - 75*e^2)*x*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(1680*c^5*e^2*sqrt(c^2*x^2)) + (b*(29*c^2*d - 25*e)*x*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(3/2))/(840*c^3*e^2*sqrt(c^2*x^2)) - (b*x*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(5/2))/(42*c*e^2*sqrt(c^2*x^2)) + (d^2*(d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*asec(c*x)))/(5*e^3) + ((d + e*x^2)^(7/2)*(a + b*asec(c*x)))/(7*e^3) + (8*b*c*d^(7/2)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(105*e^3*sqrt(c^2*x^2)) - (b*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(1680*c^6*e^(5/2)*sqrt(c^2*x^2))]
    @test_int [x^3*sqrt(d + e*x^2)*(a + b*asec(c*x)), x, If(13>=8, 11, 12), -((b*(c^2*d + 9*e)*x*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(120*c^3*e*sqrt(c^2*x^2))) - (b*x*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(3/2))/(20*c*e*sqrt(c^2*x^2)) - (d*(d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*asec(c*x)))/(5*e^2) - (2*b*c*d^(5/2)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(15*e^2*sqrt(c^2*x^2)) + (b*(15*c^4*d^2 - 10*c^2*d*e - 9*e^2)*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(120*c^4*e^(3/2)*sqrt(c^2*x^2))]
    @test_int [x^1*sqrt(d + e*x^2)*(a + b*asec(c*x)), x, 9, -((b*x*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(6*c*sqrt(c^2*x^2))) + ((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(3*e) + (b*c*d^(3/2)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(3*e*sqrt(c^2*x^2)) - (b*(3*c^2*d + e)*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(6*c^2*sqrt(e)*sqrt(c^2*x^2))]
    @test_int [sqrt(d + e*x^2)*(a + b*asec(c*x))/x^1, x, 0, Unintegrable((sqrt(d + e*x^2)*(a + b*asec(c*x)))/x, x)]
    @test_int [sqrt(d + e*x^2)*(a + b*asec(c*x))/x^3, x, 0, Unintegrable((sqrt(d + e*x^2)*(a + b*asec(c*x)))/x^3, x)]

    @test_int [x^2*sqrt(d + e*x^2)*(a + b*asec(c*x)), x, 0, Unintegrable(x^2*sqrt(d + e*x^2)*(a + b*asec(c*x)), x)]
    @test_int [x^0*sqrt(d + e*x^2)*(a + b*asec(c*x)), x, 0, Unintegrable(sqrt(d + e*x^2)*(a + b*asec(c*x)), x)]
    @test_int [sqrt(d + e*x^2)*(a + b*asec(c*x))/x^2, x, 0, Unintegrable((sqrt(d + e*x^2)*(a + b*asec(c*x)))/x^2, x)]
    @test_int [sqrt(d + e*x^2)*(a + b*asec(c*x))/x^4, x, 11, (2*b*c*(c^2*d + 2*e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(9*d*sqrt(c^2*x^2)) + (b*c*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(9*x^2*sqrt(c^2*x^2)) - ((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(3*d*x^3) - (2*b*c^2*(c^2*d + 2*e)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(9*d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*(c^2*d + e)*(2*c^2*d + 3*e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(9*d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]
    @test_int [sqrt(d + e*x^2)*(a + b*asec(c*x))/x^6, x, If(13>=8, 12, 32), If(13>=8, (b*c*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(225*d^2*sqrt(c^2*x^2)) + (b*c*(12*c^2*d - e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(225*d*x^2*sqrt(c^2*x^2)) + (b*c*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(3/2))/(25*d*x^4*sqrt(c^2*x^2)) - ((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(15*d^2*x^3) - (b*c^2*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(225*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*(c^2*d + e)*(24*c^4*d^2 + 7*c^2*d*e - 30*e^2)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(225*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)), -((2*b*c*e^2*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(15*d^2*sqrt(c^2*x^2))) + (b*c*e*(2*c^2*d + e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(45*d^2*sqrt(c^2*x^2)) + (b*c*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(75*d^2*sqrt(c^2*x^2)) + (b*c*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(25*x^4*sqrt(c^2*x^2)) + (b*c*e*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(45*d*x^2*sqrt(c^2*x^2)) + (b*c*(4*c^2*d + e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(75*d*x^2*sqrt(c^2*x^2)) - ((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(15*d^2*x^3) + (2*b*c^2*e^2*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(15*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) - (b*c^2*e*(2*c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(45*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) - (b*c^2*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(75*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*c^2*(8*c^2*d - e)*(c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(75*d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)) + (2*b*c^2*e*(c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(45*d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)) - (2*b*e^2*(c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(15*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)))]

    @test_int [x^3*(d + e*x^2)^(3/2)*(a + b*asec(c*x)), x, 12, (b*(3*c^4*d^2 - 38*c^2*d*e - 25*e^2)*x*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(560*c^5*e*sqrt(c^2*x^2)) - (b*(13*c^2*d + 25*e)*x*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(3/2))/(840*c^3*e*sqrt(c^2*x^2)) - (b*x*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(5/2))/(42*c*e*sqrt(c^2*x^2)) - (d*(d + e*x^2)^(5/2)*(a + b*asec(c*x)))/(5*e^2) + ((d + e*x^2)^(7/2)*(a + b*asec(c*x)))/(7*e^2) - (2*b*c*d^(7/2)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(35*e^2*sqrt(c^2*x^2)) + (b*(35*c^6*d^3 - 35*c^4*d^2*e - 63*c^2*d*e^2 - 25*e^3)*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(560*c^6*e^(3/2)*sqrt(c^2*x^2))]
    @test_int [x^1*(d + e*x^2)^(3/2)*(a + b*asec(c*x)), x, 10, -((b*(7*c^2*d + 3*e)*x*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(40*c^3*sqrt(c^2*x^2))) - (b*x*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(3/2))/(20*c*sqrt(c^2*x^2)) + ((d + e*x^2)^(5/2)*(a + b*asec(c*x)))/(5*e) + (b*c*d^(5/2)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(5*e*sqrt(c^2*x^2)) - (b*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(40*c^4*sqrt(e)*sqrt(c^2*x^2))]
    @test_int [(d + e*x^2)^(3/2)*(a + b*asec(c*x))/x^1, x, 0, Unintegrable(((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/x, x)]
    @test_int [(d + e*x^2)^(3/2)*(a + b*asec(c*x))/x^3, x, 0, Unintegrable(((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/x^3, x)]

    @test_int [x^2*(d + e*x^2)^(3/2)*(a + b*asec(c*x)), x, 0, Unintegrable(x^2*(d + e*x^2)^(3/2)*(a + b*asec(c*x)), x)]
    @test_int [x^0*(d + e*x^2)^(3/2)*(a + b*asec(c*x)), x, 0, Unintegrable((d + e*x^2)^(3/2)*(a + b*asec(c*x)), x)]
    @test_int [(d + e*x^2)^(3/2)*(a + b*asec(c*x))/x^2, x, 0, Unintegrable(((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/x^2, x)]
    @test_int [(d + e*x^2)^(3/2)*(a + b*asec(c*x))/x^4, x, 0, Unintegrable(((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/x^4, x)]
    @test_int [(d + e*x^2)^(3/2)*(a + b*asec(c*x))/x^6, x, 12, (b*c*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(75*d*sqrt(c^2*x^2)) + (4*b*c*(c^2*d + 2*e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(75*x^2*sqrt(c^2*x^2)) + (b*c*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(3/2))/(25*x^4*sqrt(c^2*x^2)) - ((d + e*x^2)^(5/2)*(a + b*asec(c*x)))/(5*d*x^5) - (b*c^2*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(75*d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*(c^2*d + e)*(8*c^4*d^2 + 19*c^2*d*e + 15*e^2)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(75*d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]
    @test_int [(d + e*x^2)^(3/2)*(a + b*asec(c*x))/x^8, x, 13, (b*c*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(3675*d^2*sqrt(c^2*x^2)) + (b*c*(120*c^4*d^2 + 159*c^2*d*e - 37*e^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(3675*d*x^2*sqrt(c^2*x^2)) + (b*c*(30*c^2*d + 11*e)*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(3/2))/(1225*d*x^4*sqrt(c^2*x^2)) + (b*c*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(5/2))/(49*d*x^6*sqrt(c^2*x^2)) - ((d + e*x^2)^(5/2)*(a + b*asec(c*x)))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*asec(c*x)))/(35*d^2*x^5) - (b*c^2*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(3675*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (2*b*(c^2*d + e)*(120*c^6*d^3 + 204*c^4*d^2*e + 17*c^2*d*e^2 - 105*e^3)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(3675*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5*(a + b*asec(c*x))/sqrt(d + e*x^2), x, 11, (b*(19*c^2*d - 9*e)*x*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(120*c^3*e^2*sqrt(c^2*x^2)) - (b*x*sqrt(-1 + c^2*x^2)*(d + e*x^2)^(3/2))/(20*c*e^2*sqrt(c^2*x^2)) + (d^2*sqrt(d + e*x^2)*(a + b*asec(c*x)))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*asec(c*x)))/(5*e^3) + (8*b*c*d^(5/2)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(15*e^3*sqrt(c^2*x^2)) - (b*(45*c^4*d^2 - 10*c^2*d*e + 9*e^2)*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(120*c^4*e^(5/2)*sqrt(c^2*x^2))]
    @test_int [x^3*(a + b*asec(c*x))/sqrt(d + e*x^2), x, 10, -((b*x*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(6*c*e*sqrt(c^2*x^2))) - (d*sqrt(d + e*x^2)*(a + b*asec(c*x)))/e^2 + ((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(3*e^2) - (2*b*c*d^(3/2)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(3*e^2*sqrt(c^2*x^2)) + (b*(3*c^2*d - e)*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(6*c^2*e^(3/2)*sqrt(c^2*x^2))]
    @test_int [x^1*(a + b*asec(c*x))/sqrt(d + e*x^2), x, 9, (sqrt(d + e*x^2)*(a + b*asec(c*x)))/e + (b*c*sqrt(d)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(e*sqrt(c^2*x^2)) - (b*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(sqrt(e)*sqrt(c^2*x^2))]
    @test_int [(a + b*asec(c*x))/(x^1*sqrt(d + e*x^2)), x, 0, Unintegrable((a + b*asec(c*x))/(x*sqrt(d + e*x^2)), x)]
    @test_int [(a + b*asec(c*x))/(x^3*sqrt(d + e*x^2)), x, 0, Unintegrable((a + b*asec(c*x))/(x^3*sqrt(d + e*x^2)), x)]

    @test_int [x^2*(a + b*asec(c*x))/sqrt(d + e*x^2), x, 0, Unintegrable((x^2*(a + b*asec(c*x)))/sqrt(d + e*x^2), x)]
    @test_int [x^0*(a + b*asec(c*x))/sqrt(d + e*x^2), x, 0, Unintegrable((a + b*asec(c*x))/sqrt(d + e*x^2), x)]
    @test_int [(a + b*asec(c*x))/(x^2*sqrt(d + e*x^2)), x, 11, (b*c*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(d*sqrt(c^2*x^2)) - (sqrt(d + e*x^2)*(a + b*asec(c*x)))/(d*x) - (b*c^2*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*(c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]
    @test_int [(a + b*asec(c*x))/(x^4*sqrt(d + e*x^2)), x, 11, (b*c*(2*c^2*d - 5*e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(9*d^2*sqrt(c^2*x^2)) + (b*c*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(9*d*x^2*sqrt(c^2*x^2)) - (sqrt(d + e*x^2)*(a + b*asec(c*x)))/(3*d*x^3) + (2*e*sqrt(d + e*x^2)*(a + b*asec(c*x)))/(3*d^2*x) - (b*c^2*(2*c^2*d - 5*e)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(9*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (2*b*(c^2*d - 3*e)*(c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(9*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]
    @test_int [(a + b*asec(c*x))/(x^6*sqrt(d + e*x^2)), x, 32, (8*b*c*e^2*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(15*d^3*sqrt(c^2*x^2)) - (4*b*c*e*(2*c^2*d + e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(45*d^3*sqrt(c^2*x^2)) + (b*c*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(75*d^3*sqrt(c^2*x^2)) + (b*c*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(25*d*x^4*sqrt(c^2*x^2)) - (4*b*c*e*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(45*d^2*x^2*sqrt(c^2*x^2)) + (b*c*(4*c^2*d + e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(75*d^2*x^2*sqrt(c^2*x^2)) - (sqrt(d + e*x^2)*(a + b*asec(c*x)))/(5*d*x^5) + (4*e*sqrt(d + e*x^2)*(a + b*asec(c*x)))/(15*d^2*x^3) - (8*e^2*sqrt(d + e*x^2)*(a + b*asec(c*x)))/(15*d^3*x) - (8*b*c^2*e^2*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(15*d^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (4*b*c^2*e*(2*c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(45*d^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) - (b*c^2*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(75*d^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*c^2*(8*c^2*d - e)*(c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(75*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)) - (8*b*c^2*e*(c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(45*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)) + (8*b*e^2*(c^2*d + e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(15*d^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]

    @test_int [x^5*(a + b*asec(c*x))/(d + e*x^2)^(3/2), x, 10, -((b*x*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(6*c*e^2*sqrt(c^2*x^2))) - (d^2*(a + b*asec(c*x)))/(e^3*sqrt(d + e*x^2)) - (2*d*sqrt(d + e*x^2)*(a + b*asec(c*x)))/e^3 + ((d + e*x^2)^(3/2)*(a + b*asec(c*x)))/(3*e^3) - (8*b*c*d^(3/2)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(3*e^3*sqrt(c^2*x^2)) + (b*(9*c^2*d - e)*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(6*c^2*e^(5/2)*sqrt(c^2*x^2))]
    @test_int [x^3*(a + b*asec(c*x))/(d + e*x^2)^(3/2), x, 9, (d*(a + b*asec(c*x)))/(e^2*sqrt(d + e*x^2)) + (sqrt(d + e*x^2)*(a + b*asec(c*x)))/e^2 + (2*b*c*sqrt(d)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(e^2*sqrt(c^2*x^2)) - (b*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(e^(3/2)*sqrt(c^2*x^2))]
    @test_int [x^1*(a + b*asec(c*x))/(d + e*x^2)^(3/2), x, 4, -((a + b*asec(c*x))/(e*sqrt(d + e*x^2))) - (b*c*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(sqrt(d)*e*sqrt(c^2*x^2))]
    @test_int [(a + b*asec(c*x))/(x^1*(d + e*x^2)^(3/2)), x, 0, Unintegrable((a + b*asec(c*x))/(x*(d + e*x^2)^(3/2)), x)]
    @test_int [(a + b*asec(c*x))/(x^3*(d + e*x^2)^(3/2)), x, 0, Unintegrable((a + b*asec(c*x))/(x^3*(d + e*x^2)^(3/2)), x)]

    @test_int [x^4*(a + b*asec(c*x))/(d + e*x^2)^(3/2), x, 0, Unintegrable((x^4*(a + b*asec(c*x)))/(d + e*x^2)^(3/2), x)]
    @test_int [x^2*(a + b*asec(c*x))/(d + e*x^2)^(3/2), x, 0, Unintegrable((x^2*(a + b*asec(c*x)))/(d + e*x^2)^(3/2), x)]
    @test_int [x^0*(a + b*asec(c*x))/(d + e*x^2)^(3/2), x, 5, (x*(a + b*asec(c*x)))/(d*sqrt(d + e*x^2)) - (b*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(d*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]
    @test_int [(a + b*asec(c*x))/(x^2*(d + e*x^2)^(3/2)), x, 10, (b*c*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(d^2*sqrt(c^2*x^2)) - (a + b*asec(c*x))/(d*x*sqrt(d + e*x^2)) - (2*e*x*(a + b*asec(c*x)))/(d^2*sqrt(d + e*x^2)) - (b*c^2*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*(c^2*d + 2*e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]
    @test_int [(a + b*asec(c*x))/(x^4*(d + e*x^2)^(3/2)), x, 25, (2*b*c*(c^2*d - e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(9*d^3*sqrt(c^2*x^2)) - (4*b*c*e*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(3*d^3*sqrt(c^2*x^2)) + (b*c*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(9*d^2*x^2*sqrt(c^2*x^2)) - (a + b*asec(c*x))/(3*d*x^3*sqrt(d + e*x^2)) + (4*e*(a + b*asec(c*x)))/(3*d^2*x*sqrt(d + e*x^2)) + (8*e^2*x*(a + b*asec(c*x)))/(3*d^3*sqrt(d + e*x^2)) - (2*b*c^2*(c^2*d - e)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(9*d^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (4*b*c^2*e*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(3*d^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*c^2*(2*c^2*d - e)*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(9*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)) - (4*b*c^2*e*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(3*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)) - (8*b*e^2*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(3*d^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]

    @test_int [x^5*(a + b*asec(c*x))/(d + e*x^2)^(5/2), x, 10, -((b*c*d*x*sqrt(-1 + c^2*x^2))/(3*e^2*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(d + e*x^2))) - (d^2*(a + b*asec(c*x)))/(3*e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*asec(c*x)))/(e^3*sqrt(d + e*x^2)) + (sqrt(d + e*x^2)*(a + b*asec(c*x)))/e^3 + (8*b*c*sqrt(d)*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(3*e^3*sqrt(c^2*x^2)) - (b*x*atanh((sqrt(e)*sqrt(-1 + c^2*x^2))/(c*sqrt(d + e*x^2))))/(e^(5/2)*sqrt(c^2*x^2))]
    @test_int [x^3*(a + b*asec(c*x))/(d + e*x^2)^(5/2), x, 7, (b*c*x*sqrt(-1 + c^2*x^2))/(3*e*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(d + e*x^2)) + (d*(a + b*asec(c*x)))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*asec(c*x))/(e^2*sqrt(d + e*x^2)) - (2*b*c*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(3*sqrt(d)*e^2*sqrt(c^2*x^2))]
    @test_int [x^1*(a + b*asec(c*x))/(d + e*x^2)^(5/2), x, 5, -((b*c*x*sqrt(-1 + c^2*x^2))/(3*d*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(d + e*x^2))) - (a + b*asec(c*x))/(3*e*(d + e*x^2)^(3/2)) - (b*c*x*atan(sqrt(d + e*x^2)/(sqrt(d)*sqrt(-1 + c^2*x^2))))/(3*d^(3/2)*e*sqrt(c^2*x^2))]
    @test_int [(a + b*asec(c*x))/(x^1*(d + e*x^2)^(5/2)), x, 0, Unintegrable((a + b*asec(c*x))/(x*(d + e*x^2)^(5/2)), x)]
    @test_int [(a + b*asec(c*x))/(x^3*(d + e*x^2)^(5/2)), x, 0, Unintegrable((a + b*asec(c*x))/(x^3*(d + e*x^2)^(5/2)), x)]

    @test_int [x^6*(a + b*asec(c*x))/(d + e*x^2)^(5/2), x, 0, Unintegrable((x^6*(a + b*asec(c*x)))/(d + e*x^2)^(5/2), x)]
    @test_int [x^4*(a + b*asec(c*x))/(d + e*x^2)^(5/2), x, 0, Unintegrable((x^4*(a + b*asec(c*x)))/(d + e*x^2)^(5/2), x)]
    @test_int [x^2*(a + b*asec(c*x))/(d + e*x^2)^(5/2), x, 10, -((b*c*x^2*sqrt(-1 + c^2*x^2))/(3*d*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(d + e*x^2))) + (x^3*(a + b*asec(c*x)))/(3*d*(d + e*x^2)^(3/2)) + (b*c^2*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(3*d*e*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) - (b*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(3*d*e*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]
    @test_int [x^0*(a + b*asec(c*x))/(d + e*x^2)^(5/2), x, 10, (b*c*e*x^2*sqrt(-1 + c^2*x^2))/(3*d^2*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(d + e*x^2)) + (x*(a + b*asec(c*x)))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*asec(c*x)))/(3*d^2*sqrt(d + e*x^2)) - (b*c^2*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(3*d^2*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) - (2*b*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(3*d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]
    @test_int [(a + b*asec(c*x))/(x^2*(d + e*x^2)^(5/2)), x, 26, -((b*c*e*sqrt(-1 + c^2*x^2))/(d^2*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(d + e*x^2))) - (4*b*c*e^2*x^2*sqrt(-1 + c^2*x^2))/(3*d^3*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(d + e*x^2)) + (b*c*(c^2*d + 2*e)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))/(d^3*(c^2*d + e)*sqrt(c^2*x^2)) - (a + b*asec(c*x))/(d*x*(d + e*x^2)^(3/2)) - (4*e*x*(a + b*asec(c*x)))/(3*d^2*(d + e*x^2)^(3/2)) - (8*e*x*(a + b*asec(c*x)))/(3*d^3*sqrt(d + e*x^2)) + (4*b*c^2*e*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(3*d^3*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) - (b*c^2*(c^2*d + 2*e)*x*sqrt(1 - c^2*x^2)*sqrt(d + e*x^2)*Elliptic.E(asin(c*x), -(e/(c^2*d))))/(d^3*(c^2*d + e)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(1 + (e*x^2)/d)) + (b*c^2*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(d^2*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2)) + (8*b*e*x*sqrt(1 - c^2*x^2)*sqrt(1 + (e*x^2)/d)*Elliptic.F(asin(c*x), -(e/(c^2*d))))/(3*d^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)*sqrt(d + e*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(f*x)^m*(d+e*x^2)^p*(a+b*asec(c*x))*when*m*symbolic=#

    @test_int [(f*x)^m*(d + e*x^2)^3*(a + b*asec(c*x)), x, 6, If(13>=8, -((b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*x*(f*x)^(1 + m)*sqrt(-1 + c^2*x^2))/(c^5*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*sqrt(c^2*x^2))) - (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*sqrt(-1 + c^2*x^2))/(c^3*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)*sqrt(c^2*x^2)) - (b*e^3*x*(f*x)^(5 + m)*sqrt(-1 + c^2*x^2))/(c*f^5*(6 + m)*(7 + m)*sqrt(c^2*x^2)) + (d^3*(f*x)^(1 + m)*(a + b*asec(c*x)))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*asec(c*x)))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*asec(c*x)))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*asec(c*x)))/(f^7*(7 + m)) - (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/((3 + m)*(5 + m)*(7 + m)))*x*(f*x)^(1 + m)*sqrt(1 - c^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, c^2*x^2))/(c^5*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)), -((b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*x*(f*x)^(1 + m)*sqrt(-1 + c^2*x^2))/(c^5*f*(6 + m)*(8 + 6*m + m^2)*(105 + 71*m + 15*m^2 + m^3)*sqrt(c^2*x^2))) - (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*sqrt(-1 + c^2*x^2))/(c^3*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)*sqrt(c^2*x^2)) - (b*e^3*x*(f*x)^(5 + m)*sqrt(-1 + c^2*x^2))/(c*f^5*(6 + m)*(7 + m)*sqrt(c^2*x^2)) + (d^3*(f*x)^(1 + m)*(a + b*asec(c*x)))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*asec(c*x)))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*asec(c*x)))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*asec(c*x)))/(f^7*(7 + m)) - (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/(105 + 71*m + 15*m^2 + m^3))*x*(f*x)^(1 + m)*sqrt(1 - c^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, c^2*x^2))/(c^5*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)))]
    @test_int [(f*x)^m*(d + e*x^2)^2*(a + b*asec(c*x)), x, 6, If(13>=8, -((b*e*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2))*x*(f*x)^(1 + m)*sqrt(-1 + c^2*x^2))/(c^3*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*sqrt(c^2*x^2))) - (b*e^2*x*(f*x)^(3 + m)*sqrt(-1 + c^2*x^2))/(c*f^3*(4 + m)*(5 + m)*sqrt(c^2*x^2)) + (d^2*(f*x)^(1 + m)*(a + b*asec(c*x)))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*asec(c*x)))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*asec(c*x)))/(f^5*(5 + m)) - (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2)))*x*(f*x)^(1 + m)*sqrt(1 - c^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, c^2*x^2))/(c^3*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)), -((b*e*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2))*x*(f*x)^(1 + m)*sqrt(-1 + c^2*x^2))/(c^3*f*(4 + m)*(5 + m)*(6 + 5*m + m^2)*sqrt(c^2*x^2))) - (b*e^2*x*(f*x)^(3 + m)*sqrt(-1 + c^2*x^2))/(c*f^3*(4 + m)*(5 + m)*sqrt(c^2*x^2)) + (d^2*(f*x)^(1 + m)*(a + b*asec(c*x)))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*asec(c*x)))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*asec(c*x)))/(f^5*(5 + m)) - (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2)))*x*(f*x)^(1 + m)*sqrt(1 - c^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, c^2*x^2))/(c^3*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)))]
    @test_int [(f*x)^m*(d + e*x^2)^1*(a + b*asec(c*x)), x, 5, -((b*e*x^(2 + m)*sqrt(-1 + c^2*x^2))/(c*(6 + 5*m + m^2)*sqrt(c^2*x^2))) + (d*x^(1 + m)*(a + b*asec(c*x)))/(1 + m) + (e*x^(3 + m)*(a + b*asec(c*x)))/(3 + m) + (b*(e*(1 + m)^2 + c^2*d*(2 + m)*(3 + m))*x^(2 + m)*sqrt(-1 + c^2*x^2)*HypergeometricFunctions._₂F₁(1, (2 + m)/2, (3 + m)/2, c^2*x^2))/(c*(1 + m)^2*(2 + m)*(3 + m)*sqrt(c^2*x^2)), -((b*e*x*(f*x)^(1 + m)*sqrt(-1 + c^2*x^2))/(c*f*(6 + 5*m + m^2)*sqrt(c^2*x^2))) + (d*(f*x)^(1 + m)*(a + b*asec(c*x)))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*asec(c*x)))/(f^3*(3 + m)) - (b*(e*(1 + m)^2 + c^2*d*(2 + m)*(3 + m))*x*(f*x)^(1 + m)*sqrt(1 - c^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, c^2*x^2))/(c*f*(1 + m)^2*(2 + m)*(3 + m)*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2))]
    @test_int [(f*x)^m*(a + b*asec(c*x))/(d + e*x^2)^1, x, 0, Unintegrable(((f*x)^m*(a + b*asec(c*x)))/(d + e*x^2), x)]
    @test_int [(f*x)^m*(a + b*asec(c*x))/(d + e*x^2)^2, x, 0, Unintegrable(((f*x)^m*(a + b*asec(c*x)))/(d + e*x^2)^2, x)]

    @test_int [(f*x)^m*(d + e*x^2)^(3/2)*(a + b*asec(c*x)), x, 0, Unintegrable((f*x)^m*(d + e*x^2)^(3/2)*(a + b*asec(c*x)), x)]
    @test_int [(f*x)^m*(d + e*x^2)^(1/2)*(a + b*asec(c*x)), x, 0, Unintegrable((f*x)^m*sqrt(d + e*x^2)*(a + b*asec(c*x)), x)]
    @test_int [(f*x)^m*(a + b*asec(c*x))/(d + e*x^2)^(1/2), x, 0, Unintegrable(((f*x)^m*(a + b*asec(c*x)))/sqrt(d + e*x^2), x)]
    @test_int [(f*x)^m*(a + b*asec(c*x))/(d + e*x^2)^(3/2), x, 0, Unintegrable(((f*x)^m*(a + b*asec(c*x)))/(d + e*x^2)^(3/2), x)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(f*x)^m*(d+e*x^4)^p*(a+b*asec(c*x))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x^4)^(p/2)*(a+b*asec(c*x))=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^11*(a + b*asec(c*x))/sqrt(1 - c^4*x^4), x, 16, (4*b*sqrt(1 - c^2*x^2)*sqrt(1 + c^2*x^2))/(15*c^13*sqrt(1 - 1/(c^2*x^2))*x) - (7*b*sqrt(1 - c^2*x^2)*(1 + c^2*x^2)^(3/2))/(90*c^13*sqrt(1 - 1/(c^2*x^2))*x) + (13*b*sqrt(1 - c^2*x^2)*(1 + c^2*x^2)^(5/2))/(150*c^13*sqrt(1 - 1/(c^2*x^2))*x) - (3*b*sqrt(1 - c^2*x^2)*(1 + c^2*x^2)^(7/2))/(70*c^13*sqrt(1 - 1/(c^2*x^2))*x) + (b*sqrt(1 - c^2*x^2)*(1 + c^2*x^2)^(9/2))/(90*c^13*sqrt(1 - 1/(c^2*x^2))*x) - (sqrt(1 - c^4*x^4)*(a + b*asec(c*x)))/(2*c^12) + ((1 - c^4*x^4)^(3/2)*(a + b*asec(c*x)))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*asec(c*x)))/(10*c^12) - (4*b*sqrt(1 - c^2*x^2)*atanh(sqrt(1 + c^2*x^2)))/(15*c^13*sqrt(1 - 1/(c^2*x^2))*x)]
    @test_int [x^7*(a + b*asec(c*x))/sqrt(1 - c^4*x^4), x, 13, (b*sqrt(1 - c^2*x^2)*sqrt(1 + c^2*x^2))/(3*c^9*sqrt(1 - 1/(c^2*x^2))*x) - (b*sqrt(1 - c^2*x^2)*(1 + c^2*x^2)^(3/2))/(18*c^9*sqrt(1 - 1/(c^2*x^2))*x) + (b*sqrt(1 - c^2*x^2)*(1 + c^2*x^2)^(5/2))/(30*c^9*sqrt(1 - 1/(c^2*x^2))*x) - (sqrt(1 - c^4*x^4)*(a + b*asec(c*x)))/(2*c^8) + ((1 - c^4*x^4)^(3/2)*(a + b*asec(c*x)))/(6*c^8) - (b*sqrt(1 - c^2*x^2)*atanh(sqrt(1 + c^2*x^2)))/(3*c^9*sqrt(1 - 1/(c^2*x^2))*x)]
    @test_int [x^3*(a + b*asec(c*x))/sqrt(1 - c^4*x^4), x, 8, (b*x*sqrt(1 - c^4*x^4))/(2*c^3*sqrt(c^2*x^2)*sqrt(-1 + c^2*x^2)) - (sqrt(1 - c^4*x^4)*(a + b*asec(c*x)))/(2*c^4) - (b*x*atan(sqrt(1 - c^4*x^4)/sqrt(-1 + c^2*x^2)))/(2*c^3*sqrt(c^2*x^2)), (b*sqrt(1 - c^2*x^2)*sqrt(1 + c^2*x^2))/(2*c^5*sqrt(1 - 1/(c^2*x^2))*x) - (sqrt(1 - c^4*x^4)*(a + b*asec(c*x)))/(2*c^4) - (b*sqrt(1 - c^2*x^2)*atanh(sqrt(1 + c^2*x^2)))/(2*c^5*sqrt(1 - 1/(c^2*x^2))*x)]
    @test_int [(a + b*asec(c*x))/(x^1*sqrt(1 - c^4*x^4)), x, 0, Unintegrable((a + b*asec(c*x))/(x*sqrt(1 - c^4*x^4)), x)]
    @test_int [(a + b*asec(c*x))/(x^5*sqrt(1 - c^4*x^4)), x, 0, Unintegrable((a + b*asec(c*x))/(x^5*sqrt(1 - c^4*x^4)), x)]

    #= ::Section:: =#
    #=Integrands*of*the*form*u*(a+b*asec(c*x))^n=#
    end

max_stars_repo_path: test/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.jl


text:
using Plots
using Convex
using Mosek, MosekTools
using SCS
using DelimitedFiles
using SparseArrays

data = readdlm(joinpath(dirname(@__FILE__),"data","snp500.txt"))
n = size(data)[1]

#p = plot(1:length(data), data)

N = length(data)
H =  spdiagm(0=>ones(N-2), 1=>-2*ones(N-2), 2=>ones(N-2))

x = Variable(N)

λ = 50
problem = minimize(0.5*sumsquares(x - data) + λ*norm(H*x, 1))

# Solve the problem by calling solve!
solve!(problem, Mosek.Optimizer)
solve!(problem, SCS.Optimizer)

# Check the status of the problem
problem.status # :Optimal, :Infeasible, :Unbounded etc.

# Get the optimal value
problem.optval

plot(data)
plot!(evaluate(x))

ext: jl