text:
using Rubin
using Tests
using Elliptic
using HypergeometricFunctions
using Polylogarithms
using SpecialFunctions

@test integrate((a+a*sin(e+f*x))^2*sin(e+f*x)^3, x) == :(3//4*x*a^2+a^2*f^-1*cos(e+f*x)^3+-2*a^2*f^-1*cos(e+f*x)+-1//5*a^2*f^-1*cos(e+f*x)^5+-3//4*a^2*f^-1*cos(e+f*x)*sin(e+f*x)+-1//2*a^2*f^-1*sin(e+f*x)^3*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*sin(e+f*x)^3, x) == :(23//16*x*a^3+-4*a^3*f^-1*cos(e+f*x)+-3//5*a^3*f^-1*cos(e+f*x)^5+7//3*a^3*f^-1*cos(e+f*x)^3+-23//16*a^3*f^-1*cos(e+f*x)*sin(e+f*x)+-23//24*a^3*f^-1*sin(e+f*x)^3*cos(e+f*x)+-1//6*a^3*f^-1*sin(e+f*x)^5*cos(e+f*x))
@test integrate((a+a*sin(x))^-1*sin(x)^4, x) == :(-4*a^-1*cos(x)+-3//2*x*a^-1+4//3*a^-1*cos(x)^3+(a+a*sin(x))^-1*sin(x)^3*cos(x)+3//2*a^-1*cos(x)*sin(x))
@test integrate((a+a*sin(x))^-1*sin(x)^3, x) == :(2*a^-1*cos(x)+3//2*x*a^-1+(a+a*sin(x))^-1*sin(x)^2*cos(x)+-3//2*a^-1*cos(x)*sin(x))
@test integrate((a+a*sin(x))^-1*sin(x)^2, x) == :(-1*x*a^-1+-1*a^-1*cos(x)+-1*a^-1*(1+sin(x))^-1*cos(x))
@test integrate((a+a*sin(x))^-1*sin(x), x) == :(x*a^-1+(a+a*sin(x))^-1*cos(x))
@test integrate((a+a*sin(x))^-1, x) == :(-1*(a+a*sin(x))^-1*cos(x))
@test integrate((a+a*sin(x))^-1*csc(x), x) == :((a+a*sin(x))^-1*cos(x)+-1*a^-1*arctanh(cos(x)))
@test integrate((a+a*sin(x))^-1*csc(x)^2, x) == :(a^-1*arctanh(cos(x))+(a+a*sin(x))^-1*cot(x)+-2*a^-1*cot(x))
@test integrate((a+a*sin(x))^-1*csc(x)^3, x) == :(2*a^-1*cot(x)+-3//2*a^-1*arctanh(cos(x))+(a+a*sin(x))^-1*cot(x)*csc(x)+-3//2*a^-1*cot(x)*csc(x))
@test integrate((a+a*sin(x))^-1*csc(x)^4, x) == :(-4*a^-1*cot(x)+-4//3*a^-1*cot(x)^3+3//2*a^-1*arctanh(cos(x))+(a+a*sin(x))^-1*csc(x)^2*cot(x)+3//2*a^-1*cot(x)*csc(x))
@test integrate((a+a*sin(x))^-2*sin(x)^4, x) == :(7//2*x*a^-2+16//3*a^-2*cos(x)+-7//2*a^-2*cos(x)*sin(x)+1//3*(a+a*sin(x))^-2*sin(x)^3*cos(x)+8//3*a^-2*(1+sin(x))^-1*sin(x)^2*cos(x))
@test integrate((a+a*sin(x))^-2*sin(x)^3, x) == :(-2*x*a^-2+-4//3*a^-2*cos(x)+-2*a^-2*(1+sin(x))^-1*cos(x)+1//3*(a+a*sin(x))^-2*sin(x)^2*cos(x))
@test integrate((a+a*sin(x))^-2*sin(x)^2, x) == :(x*a^-2+-1//3*(a+a*sin(x))^-2*cos(x)+5//3*a^-2*(1+sin(x))^-1*cos(x))
@test integrate((a+a*sin(x))^-2*sin(x), x) == :(-2*(3*a^2+3*a^2*sin(x))^-1*cos(x)+1//3*(a+a*sin(x))^-2*cos(x))
@test integrate((a+a*sin(x))^-2, x) == :(-1*(3*a^2+3*a^2*sin(x))^-1*cos(x)+-1//3*(a+a*sin(x))^-2*cos(x))
@test integrate((a+a*sin(x))^-2*csc(x), x) == :(-1*a^-2*arctanh(cos(x))+1//3*(a+a*sin(x))^-2*cos(x)+4//3*a^-2*(1+sin(x))^-1*cos(x))
@test integrate((a+a*sin(x))^-2*csc(x)^2, x) == :(2*a^-2*arctanh(cos(x))+-10//3*a^-2*cot(x)+1//3*(a+a*sin(x))^-2*cot(x)+2*a^-2*(1+sin(x))^-1*cot(x))
@test integrate((a+a*sin(x))^-2*csc(x)^3, x) == :(-7//2*a^-2*arctanh(cos(x))+16//3*a^-2*cot(x)+-7//2*a^-2*cot(x)*csc(x)+1//3*(a+a*sin(x))^-2*cot(x)*csc(x)+8//3*a^-2*(1+sin(x))^-1*cot(x)*csc(x))
@test integrate((a+a*sin(x))^-3*sin(x)^6, x) == :(-136//5*a^-3*cos(x)+-23//2*x*a^-3+136//15*a^-3*cos(x)^3+23*(3*a^3+3*a^3*sin(x))^-1*sin(x)^3*cos(x)+1//5*(a+a*sin(x))^-3*sin(x)^5*cos(x)+23//2*a^-3*cos(x)*sin(x)+13//15*a^-1*(a+a*sin(x))^-2*sin(x)^4*cos(x))
@test integrate((a+a*sin(x))^-3*sin(x)^5, x) == :(13//2*x*a^-3+152//15*a^-3*cos(x)+76*(15*a^3+15*a^3*sin(x))^-1*sin(x)^2*cos(x)+-13//2*a^-3*cos(x)*sin(x)+1//5*(a+a*sin(x))^-3*sin(x)^4*cos(x)+11//15*a^-1*(a+a*sin(x))^-2*sin(x)^3*cos(x))
@test integrate((a+a*sin(x))^-3*sin(x)^4, x) == :(-3*x*a^-3+-3*(a^3+a^3*sin(x))^-1*cos(x)+-9//5*a^-3*cos(x)+1//5*(a+a*sin(x))^-3*sin(x)^3*cos(x)+3//5*a^-1*(a+a*sin(x))^-2*sin(x)^2*cos(x))
@test integrate((a+a*sin(x))^-3*sin(x)^3, x) == :(x*a^-3+29*(15*a^3+15*a^3*sin(x))^-1*cos(x)+-7//15*a^-1*(a+a*sin(x))^-2*cos(x)+1//5*(a+a*sin(x))^-3*sin(x)^2*cos(x))
@test integrate((a+a*sin(x))^-3*sin(x)^2, x) == :(-7*(15*a^3+15*a^3*sin(x))^-1*cos(x)+-1//5*(a+a*sin(x))^-3*cos(x)+8//15*a^-1*(a+a*sin(x))^-2*cos(x))
@test integrate((a+a*sin(x))^-3*sin(x), x) == :(-1*(5*a^3+5*a^3*sin(x))^-1*cos(x)+1//5*(a+a*sin(x))^-3*cos(x)+-1//5*a^-1*(a+a*sin(x))^-2*cos(x))
@test integrate((a+a*sin(x))^-3, x) == :(-2*(15*a^3+15*a^3*sin(x))^-1*cos(x)+-1//5*(a+a*sin(x))^-3*cos(x)+-2//15*a^-1*(a+a*sin(x))^-2*cos(x))
@test integrate((a+a*sin(x))^-3*csc(x), x) == :(-1*a^-3*arctanh(cos(x))+22*(15*a^3+15*a^3*sin(x))^-1*cos(x)+1//5*(a+a*sin(x))^-3*cos(x)+7//15*a^-1*(a+a*sin(x))^-2*cos(x))
@test integrate((a+a*sin(x))^-3*csc(x)^2, x) == :(3*a^-3*arctanh(cos(x))+3*(a^3+a^3*sin(x))^-1*cot(x)+-24//5*a^-3*cot(x)+1//5*(a+a*sin(x))^-3*cot(x)+3//5*a^-1*(a+a*sin(x))^-2*cot(x))
@test integrate((a+a*sin(x))^-3*csc(x)^3, x) == :(-13//2*a^-3*arctanh(cos(x))+152//15*a^-3*cot(x)+76*(15*a^3+15*a^3*sin(x))^-1*cot(x)*csc(x)+-13//2*a^-3*cot(x)*csc(x)+1//5*(a+a*sin(x))^-3*cot(x)*csc(x)+11//15*a^-1*(a+a*sin(x))^-2*cot(x)*csc(x))
@test integrate((a+a*sin(x))^-3*csc(x)^4, x) == :(-136//5*a^-3*cot(x)+-136//15*a^-3*cot(x)^3+23//2*a^-3*arctanh(cos(x))+23*(3*a^3+3*a^3*sin(x))^-1*csc(x)^2*cot(x)+1//5*(a+a*sin(x))^-3*csc(x)^2*cot(x)+23//2*a^-3*cot(x)*csc(x)+13//15*a^-1*(a+a*sin(x))^-2*csc(x)^2*cot(x))
@test integrate((a+a*sin(c+d*x))^(1/2)*sin(c+d*x)^4, x) == :(64//315*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-32//45*a*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-32//105*a^-1*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)+-16//63*a*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^3*cos(c+d*x)+-2//9*a*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^4*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^(1/2)*sin(c+d*x)^3, x) == :(8//35*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-12//35*a^-1*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)+-4//5*a*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-2//7*a*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^3*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^(1/2)*sin(c+d*x)^2, x) == :(4//15*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-14//15*a*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-2//5*a^-1*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^(1/2)*sin(c+d*x), x) == :(-2//3*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-2//3*a*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^(1/2), x) == :(-2*a*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^(1/2)*csc(c+d*x), x) == :(-2*a^(1/2)*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^(1/2)*csc(c+d*x)^2, x) == :(-1*a^(1/2)*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-1*a*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x))
@test integrate((a+a*sin(c+d*x))^(1/2)*csc(c+d*x)^3, x) == :(-3//4*a^(1/2)*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-3//4*a*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+-1//2*a*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x))
@test integrate((a+a*sin(c+d*x))^(1/2)*csc(c+d*x)^4, x) == :(-5//8*a^(1/2)*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-5//8*a*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+-5//12*a*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x)+-1//3*a*d^-1*(a+a*sin(c+d*x))^-1//2*csc(c+d*x)^2*cot(c+d*x))
@test integrate((a+-1*a*sin(c+d*x))^(1/2)*csc(c+d*x), x) == :(-2*a^(1/2)*d^-1*arctanh(a^(1/2)*(a+-1*a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((-1a+a*sin(c+d*x))^(1/2)*csc(c+d*x), x) == :(2*a^(1/2)*d^-1*arctan(a^(1/2)*(-1a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((-1a+-1*a*sin(c+d*x))^(1/2)*csc(c+d*x), x) == :(2*a^(1/2)*d^-1*arctan(a^(1/2)*(-1a+-1*a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^3//2*sin(c+d*x)^3, x) == :(-68//105*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)+-68//45*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+136//315*a*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-34//63*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^3*cos(c+d*x)+-2//9*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^4*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2*sin(c+d*x)^2, x) == :(4//35*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)+-152//105*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-38//105*a*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-2//7*a^-1*d^-1*(a+a*sin(c+d*x))^5//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2*sin(c+d*x), x) == :(-2//5*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)+-8//5*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-2//5*a*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2, x) == :(-8//3*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-2//3*a*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2*csc(c+d*x), x) == :(-2*a^3//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-2*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2*csc(c+d*x)^2, x) == :(-3*a^3//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-1*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2*csc(c+d*x)^3, x) == :(-7//4*a^3//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-7//4*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+-1//2*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x))
@test integrate((a+a*sin(c+d*x))^3//2*csc(c+d*x)^4, x) == :(-11//8*a^3//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-11//8*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+-11//12*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x)+-1//3*a^2*d^-1*(a+a*sin(c+d*x))^-1//2*csc(c+d*x)^2*cot(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*sin(c+d*x)^3, x) == :(-284//99*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-284//231*a*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)+568//693*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-710//693*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^3*cos(c+d*x)+-46//99*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^4*cos(c+d*x)+-2//11*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*sin(c+d*x)^4*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*sin(c+d*x)^2, x) == :(4//63*d^-1*(a+a*sin(c+d*x))^5//2*cos(c+d*x)+-832//315*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-208//315*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-26//105*a*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x)+-2//9*a^-1*d^-1*(a+a*sin(c+d*x))^7//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*sin(c+d*x), x) == :(-2//7*d^-1*(a+a*sin(c+d*x))^5//2*cos(c+d*x)+-64//21*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-16//21*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-2//7*a*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2, x) == :(-64//15*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-16//15*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-2//5*a*d^-1*(a+a*sin(c+d*x))^3//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*csc(c+d*x), x) == :(-2*a^5//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-14//3*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-2//3*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*csc(c+d*x)^2, x) == :(-5*a^5//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-1*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cot(c+d*x)+-1*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*csc(c+d*x)^3, x) == :(-19//4*a^5//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-9//4*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+-1//2*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*cot(c+d*x)*csc(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*csc(c+d*x)^4, x) == :(-25//8*a^5//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-25//8*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+-13//12*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x)+-1//3*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*csc(c+d*x)^2*cot(c+d*x))
@test integrate((a+a*sin(c+d*x))^5//2*csc(c+d*x)^5, x) == :(-163//64*a^5//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-163//64*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+-163//96*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x)+-17//24*a^3*d^-1*(a+a*sin(c+d*x))^-1//2*csc(c+d*x)^2*cot(c+d*x)+-1//4*a^2*d^-1*(a+a*sin(c+d*x))^(1/2)*csc(c+d*x)^3*cot(c+d*x))
@test integrate((a+a*sin(c+d*x))^-1//2*sin(c+d*x)^3, x) == :(-28//15*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-2//5*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^2*cos(c+d*x)+2//15*a^-1*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-1//2*sin(c+d*x)^2, x) == :(4//3*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+-1*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-2//3*a^-1*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-1//2*sin(c+d*x), x) == :(-2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-1//2, x) == :(-1*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-1//2*csc(c+d*x), x) == :(-2*a^-1//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-1//2*csc(c+d*x)^2, x) == :(a^-1//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-1*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+-1*2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-1//2*csc(c+d*x)^3, x) == :(-7//4*a^-1//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+1//4*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+2^(1/2)*a^-1//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-1//2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3//2*sin(c+d*x)^4, x) == :((1/2)*d^-1*(a+a*sin(c+d*x))^-3//2*sin(c+d*x)^3*cos(c+d*x)+-31//5*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+13//10*a^-2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+15//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-9//10*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3//2*sin(c+d*x)^3, x) == :((1/2)*d^-1*(a+a*sin(c+d*x))^-3//2*sin(c+d*x)^2*cos(c+d*x)+-11//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-7//6*a^-2*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+13//3*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3//2*sin(c+d*x)^2, x) == :(-1//2*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-2*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+7//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3//2*sin(c+d*x), x) == :((1/2)*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-3//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3//2, x) == :(-1//2*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-1//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3//2*csc(c+d*x), x) == :((1/2)*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-2*a^-3//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+5//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-3//2*csc(c+d*x)^2, x) == :((1/2)*d^-1*(a+a*sin(c+d*x))^-3//2*cot(c+d*x)+3*a^-3//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-9//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-3//2*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x))
@test integrate((a+a*sin(c+d*x))^-3//2*csc(c+d*x)^3, x) == :(-19//4*a^-3//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+(1/2)*d^-1*(a+a*sin(c+d*x))^-3//2*cot(c+d*x)*csc(c+d*x)+7//4*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+13//4*2^(1/2)*a^-3//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-1*a^-1*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x))
@test integrate((a+a*sin(c+d*x))^-5//2*sin(c+d*x)^5, x) == :(-1729//120*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+1//4*d^-1*(a+a*sin(c+d*x))^-5//2*sin(c+d*x)^4*cos(c+d*x)+283//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+787//240*a^-3*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+-157//80*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*sin(c+d*x)^2*cos(c+d*x)+21//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*sin(c+d*x)^3*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-5//2*sin(c+d*x)^4, x) == :(-163//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-95//48*a^-3*d^-1*(a+a*sin(c+d*x))^(1/2)*cos(c+d*x)+1//4*d^-1*(a+a*sin(c+d*x))^-5//2*sin(c+d*x)^3*cos(c+d*x)+197//24*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+17//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*sin(c+d*x)^2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-5//2*sin(c+d*x)^3, x) == :(-13//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-9//4*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)+1//4*d^-1*(a+a*sin(c+d*x))^-5//2*sin(c+d*x)^2*cos(c+d*x)+75//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-5//2*sin(c+d*x)^2, x) == :(-1//4*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)+-19//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+13//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x))
@test integrate((a+a*sin(c+d*x))^-5//2*sin(c+d*x), x) == :(1//4*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)+-5//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-5//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-5//2, x) == :(-1//4*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)+-3//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+-3//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-5//2*csc(c+d*x), x) == :(-2*a^-5//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+1//4*d^-1*(a+a*sin(c+d*x))^-5//2*cos(c+d*x)+11//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cos(c+d*x)+43//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x)))
@test integrate((a+a*sin(c+d*x))^-5//2*csc(c+d*x)^2, x) == :(5*a^-5//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+1//4*d^-1*(a+a*sin(c+d*x))^-5//2*cot(c+d*x)+-115//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-35//16*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+15//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cot(c+d*x))
@test integrate((a+a*sin(c+d*x))^-5//2*csc(c+d*x)^3, x) == :(-39//4*a^-5//2*d^-1*arctanh(a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+1//4*d^-1*(a+a*sin(c+d*x))^-5//2*cot(c+d*x)*csc(c+d*x)+63//16*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)+219//32*2^(1/2)*a^-5//2*d^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(c+d*x))^-1//2*cos(c+d*x))+-31//16*a^-2*d^-1*(a+a*sin(c+d*x))^-1//2*cot(c+d*x)*csc(c+d*x)+19//16*a^-1*d^-1*(a+a*sin(c+d*x))^-3//2*cot(c+d*x)*csc(c+d*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*sin(e+f*x)^-1//2, x) == :(-2*a^(1/2)*f^-1*arcsin(a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((-1*sin(e+f*x))^-1//2*(a+-1*a*sin(e+f*x))^(1/2), x) == :(2*a^(1/2)*f^-1*arcsin(a^(1/2)*(a+-1*a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((1+sin(x))^-1//2*sin(x)^-1//2, x) == :(-1*2^(1/2)*arcsin((1+sin(x))^-1*cos(x)))
@test integrate((a+a*sin(x))^-1//2*sin(x)^-1//2, x) == :(-1*2^(1/2)*a^-1//2*arctan((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(x))^-1//2*sin(x)^-1//2*cos(x)))
@test integrate((1+-1*sin(x))^-1//2*sin(x)^-1//2, x) == :(2^(1/2)*arctanh((1/2)*2^(1/2)*(1+-1*sin(x))^-1//2*sin(x)^-1//2*cos(x)))
@test integrate((a+-1*a*sin(x))^-1//2*sin(x)^-1//2, x) == :(2^(1/2)*a^-1//2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+-1*a*sin(x))^-1//2*sin(x)^-1//2*cos(x)))
@test integrate(sin(e+f*x)^3*(a+b*sin(e+f*x)), x) == :(3//8*b*x+-1*a*f^-1*cos(e+f*x)+1//3*a*f^-1*cos(e+f*x)^3+-3//8*b*f^-1*cos(e+f*x)*sin(e+f*x)+-1//4*b*f^-1*sin(e+f*x)^3*cos(e+f*x))
@test integrate(sin(e+f*x)^2*(a+b*sin(e+f*x)), x) == :((1/2)*a*x+-1*b*f^-1*cos(e+f*x)+1//3*b*f^-1*cos(e+f*x)^3+-1//2*a*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))*sin(e+f*x), x) == :((1/2)*b*x+-1*a*f^-1*cos(e+f*x)+-1//2*b*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate(a+b*sin(e+f*x), x) == :(a*x+-1*b*f^-1*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))*csc(e+f*x), x) == :(b*x+-1*a*f^-1*arctanh(cos(e+f*x)))
@test integrate(csc(e+f*x)^2*(a+b*sin(e+f*x)), x) == :(-1*a*f^-1*cot(e+f*x)+-1*b*f^-1*arctanh(cos(e+f*x)))
@test integrate(csc(e+f*x)^3*(a+b*sin(e+f*x)), x) == :(-1*b*f^-1*cot(e+f*x)+-1//2*a*f^-1*arctanh(cos(e+f*x))+-1//2*a*f^-1*cot(e+f*x)*csc(e+f*x))
@test integrate(csc(e+f*x)^4*(a+b*sin(e+f*x)), x) == :(-1*a*f^-1*cot(e+f*x)+-1//2*b*f^-1*arctanh(cos(e+f*x))+-1//3*a*f^-1*cot(e+f*x)^3+-1//2*b*f^-1*cot(e+f*x)*csc(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*sin(e+f*x)^3, x) == :(-1*f^-1*(a^2+b^2)*cos(e+f*x)+-1//5*b^2*f^-1*cos(e+f*x)^5+1//3*f^-1*cos(e+f*x)^3*(a^2+2*b^2)+3//4*a*b*x+-3//4*a*b*f^-1*cos(e+f*x)*sin(e+f*x)+-1//2*a*b*f^-1*sin(e+f*x)^3*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*sin(e+f*x)^2, x) == :(x*((1/2)*a^2+3//8*b^2)+-2*a*b*f^-1*cos(e+f*x)+-1//4*b^2*f^-1*sin(e+f*x)^3*cos(e+f*x)+-1//8*f^-1*(3*b^2+4*a^2)*cos(e+f*x)*sin(e+f*x)+2//3*a*b*f^-1*cos(e+f*x)^3)
@test integrate((a+b*sin(e+f*x))^2*sin(e+f*x), x) == :(a*b*x+-1//3*f^-1*(a+b*sin(e+f*x))^2*cos(e+f*x)+-1//3*f^-1*(2*a^2+2*b^2)*cos(e+f*x)+-1//3*a*b*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^2, x) == :(x*(a^2+(1/2)*b^2)+-2*a*b*f^-1*cos(e+f*x)+-1//2*b^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*csc(e+f*x), x) == :(-1*a^2*f^-1*arctanh(cos(e+f*x))+-1*b^2*f^-1*cos(e+f*x)+2*a*b*x)
@test integrate((a+b*sin(e+f*x))^2*csc(e+f*x)^2, x) == :(x*b^2+-1*a^2*f^-1*cot(e+f*x)+-2*a*b*f^-1*arctanh(cos(e+f*x)))
@test integrate((a+b*sin(e+f*x))^2*csc(e+f*x)^3, x) == :(-1//2*f^-1*(a^2+2*b^2)*arctanh(cos(e+f*x))+-2*a*b*f^-1*cot(e+f*x)+-1//2*a^2*f^-1*cot(e+f*x)*csc(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*csc(e+f*x)^4, x) == :(-1//3*f^-1*(2*a^2+3*b^2)*cot(e+f*x)+-1*a*b*f^-1*arctanh(cos(e+f*x))+-1//3*a^2*f^-1*csc(e+f*x)^2*cot(e+f*x)+-1*a*b*f^-1*cot(e+f*x)*csc(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*csc(e+f*x)^5, x) == :(-1//8*f^-1*(3*a^2+4*b^2)*arctanh(cos(e+f*x))+-2*a*b*f^-1*cot(e+f*x)+-2//3*a*b*f^-1*cot(e+f*x)^3+-1//4*a^2*f^-1*csc(e+f*x)^3*cot(e+f*x)+-1//8*f^-1*(3*a^2+4*b^2)*cot(e+f*x)*csc(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*sin(e+f*x), x) == :(-1//4*f^-1*(a+b*sin(e+f*x))^3*cos(e+f*x)+3//8*b*x*(b^2+4*a^2)+-1//2*a*f^-1*(a^2+4*b^2)*cos(e+f*x)+-1//4*a*f^-1*(a+b*sin(e+f*x))^2*cos(e+f*x)+-1//8*b*f^-1*(2*a^2+3*b^2)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^3, x) == :((1/2)*a*x*(2*a^2+3*b^2)+-2//3*b*f^-1*(b^2+4*a^2)*cos(e+f*x)+-1//3*b*f^-1*(a+b*sin(e+f*x))^2*cos(e+f*x)+-5//6*a*b^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*csc(e+f*x), x) == :((1/2)*b*x*(b^2+6*a^2)+-1*a^3*f^-1*arctanh(cos(e+f*x))+-5//2*a*b^2*f^-1*cos(e+f*x)+-1//2*b^2*f^-1*(a+b*sin(e+f*x))*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*csc(e+f*x)^2, x) == :(3*a*x*b^2+b*f^-1*(a^2+-1*b^2)*cos(e+f*x)+-1*a^2*f^-1*(a+b*sin(e+f*x))*cot(e+f*x)+-3*b*a^2*f^-1*arctanh(cos(e+f*x)))
@test integrate((a+b*sin(e+f*x))^3*csc(e+f*x)^3, x) == :(x*b^3+-5//2*b*a^2*f^-1*cot(e+f*x)+-1//2*a*f^-1*(a^2+6*b^2)*arctanh(cos(e+f*x))+-1//2*a^2*f^-1*(a+b*sin(e+f*x))*cot(e+f*x)*csc(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*csc(e+f*x)^4, x) == :(-1//2*b*f^-1*(2*b^2+3*a^2)*arctanh(cos(e+f*x))+-1//3*a*f^-1*(2*a^2+9*b^2)*cot(e+f*x)+-7//6*b*a^2*f^-1*cot(e+f*x)*csc(e+f*x)+-1//3*a^2*f^-1*csc(e+f*x)^2*(a+b*sin(e+f*x))*cot(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*csc(e+f*x)^5, x) == :(-1*b*f^-1*(b^2+2*a^2)*cot(e+f*x)+-3//8*a*f^-1*(a^2+4*b^2)*arctanh(cos(e+f*x))+-3//4*b*a^2*f^-1*csc(e+f*x)^2*cot(e+f*x)+-3//8*a*f^-1*(a^2+4*b^2)*cot(e+f*x)*csc(e+f*x)+-1//4*a^2*f^-1*csc(e+f*x)^3*(a+b*sin(e+f*x))*cot(e+f*x))
@test integrate((a+b*sin(e+f*x))^4, x) == :(x*(a^4+3//8*b^4+3*a^2*b^2)+-1//4*b*f^-1*(a+b*sin(e+f*x))^3*cos(e+f*x)+-7//12*a*b*f^-1*(a+b*sin(e+f*x))^2*cos(e+f*x)+-1//6*a*b*f^-1*(16*b^2+19*a^2)*cos(e+f*x)+-1//24*b^2*f^-1*(9*b^2+26*a^2)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(x))^-1*sin(x)^4, x) == :(-1//3*b^-1*sin(x)^2*cos(x)+-1//3*b^-3*(2*b^2+3*a^2)*cos(x)+(1/2)*a*b^-2*cos(x)*sin(x)+2*a^4*b^-4*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1//2*a*x*b^-4*(b^2+2*a^2))
@test integrate((a+b*sin(x))^-1*sin(x)^3, x) == :(a*b^-2*cos(x)+(1/2)*x*b^-3*(b^2+2*a^2)+-1//2*b^-1*cos(x)*sin(x)+-2*a^3*b^-3*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-1*sin(x)^2, x) == :(-1*b^-1*cos(x)+-1*a*x*b^-2+2*a^2*b^-2*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-1*sin(x), x) == :(x*b^-1+-2*a*b^-1*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-1, x) == :(2*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-1*csc(x), x) == :(-1*a^-1*arctanh(cos(x))+-2*b*a^-1*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-1*csc(x)^2, x) == :(-1*a^-1*cot(x)+b*a^-2*arctanh(cos(x))+2*a^-2*b^2*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-1*csc(x)^3, x) == :(b*a^-2*cot(x)+-1//2*a^-1*cot(x)*csc(x)+-1//2*a^-3*(a^2+2*b^2)*arctanh(cos(x))+-2*a^-3*b^3*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-1*csc(x)^4, x) == :(-1//3*a^-1*csc(x)^2*cot(x)+-1//3*a^-3*(2*a^2+3*b^2)*cot(x)+(1/2)*b*a^-4*(a^2+2*b^2)*arctanh(cos(x))+(1/2)*b*a^-2*cot(x)*csc(x)+2*a^-4*b^4*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-2*sin(x)^4, x) == :((1/2)*x*b^-4*(b^2+6*a^2)+a*b^-3*(a^2+-1*b^2)^-1*(-2*b^2+3*a^2)*cos(x)+-2*a^3*b^-4*(a^2+-1*b^2)^-3//2*(-4*b^2+3*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1//2*b^-2*(a^2+-1*b^2)^-1*(-1*b^2+3*a^2)*cos(x)*sin(x)+a^2*b^-1*(a+b*sin(x))^-1*(a^2+-1*b^2)^-1*sin(x)^2*cos(x))
@test integrate((a+b*sin(x))^-2*sin(x)^3, x) == :(-2*a*x*b^-3+-1*b^-2*(a^2+-1*b^2)^-1*(-1*b^2+2*a^2)*cos(x)+2*a^2*b^-3*(a^2+-1*b^2)^-3//2*(-3*b^2+2*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+a^2*b^-1*(a+b*sin(x))^-1*(a^2+-1*b^2)^-1*cos(x)*sin(x))
@test integrate((a+b*sin(x))^-2*sin(x)^2, x) == :(x*b^-2+a^2*b^-1*(a+b*sin(x))^-1*(a^2+-1*b^2)^-1*cos(x)+-2*a*b^-2*(a^2+-1*b^2)^-3//2*(a^2+-2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-2*sin(x), x) == :(-2*b*(a^2+-1*b^2)^-3//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1*a*(a+b*sin(x))^-1*(a^2+-1*b^2)^-1*cos(x))
@test integrate((a+b*sin(x))^-2, x) == :(2*a*(a^2+-1*b^2)^-3//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+b*(a+b*sin(x))^-1*(a^2+-1*b^2)^-1*cos(x))
@test integrate((a+b*sin(x))^-2*csc(x), x) == :(-1*a^-2*arctanh(cos(x))+-1*a^-1*b^2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-1*cos(x)+-2*b*a^-2*(a^2+-1*b^2)^-3//2*(-1*b^2+2*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-2*csc(x)^2, x) == :(2*b*a^-3*arctanh(cos(x))+-1*a^-2*(a^2+-1*b^2)^-1*(a^2+-2*b^2)*cot(x)+-1*a^-1*b^2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-1*cot(x)+2*a^-3*b^2*(a^2+-1*b^2)^-3//2*(-2*b^2+3*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-2*csc(x)^3, x) == :(-1//2*a^-4*(a^2+6*b^2)*arctanh(cos(x))+b*a^-3*(a^2+-1*b^2)^-1*(-3*b^2+2*a^2)*cot(x)+-2*a^-4*b^3*(a^2+-1*b^2)^-3//2*(-3*b^2+4*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1//2*a^-2*(a^2+-1*b^2)^-1*(a^2+-3*b^2)*cot(x)*csc(x)+-1*a^-1*b^2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-1*cot(x)*csc(x))
@test integrate((a+b*sin(x))^-3*sin(x)^5, x) == :((1/2)*x*b^-5*(b^2+12*a^2)+-1*a^3*b^-5*(a^2+-1*b^2)^-5//2*(12*a^4+20*b^4+-29*a^2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1//2*b^-3*(a^2+-1*b^2)^-2*(b^4+6*a^4+-10*a^2*b^2)*cos(x)*sin(x)+3//2*a*b^-4*(a^2+-1*b^2)^-2*(2*b^4+4*a^4+-7*a^2*b^2)*cos(x)+(1/2)*a^2*b^-1*(a+b*sin(x))^-2*(a^2+-1*b^2)^-1*sin(x)^3*cos(x)+(1/2)*a^2*b^-2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*sin(x)^2*(-7*b^2+4*a^2)*cos(x))
@test integrate((a+b*sin(x))^-3*sin(x)^4, x) == :(-3*a*x*b^-4+-1//2*b^-3*(a^2+-1*b^2)^-1*(-2*b^2+3*a^2)*cos(x)+3*a^2*b^-4*(a^2+-1*b^2)^-5//2*(2*a^4+4*b^4+-5*a^2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+(1/2)*a^2*b^-1*(a+b*sin(x))^-2*(a^2+-1*b^2)^-1*sin(x)^2*cos(x)+-3//2*a^3*b^-3*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*(a^2+-2*b^2)*cos(x))
@test integrate((a+b*sin(x))^-3*sin(x)^3, x) == :(x*b^-3+-1*a*b^-3*(a^2+-1*b^2)^-5//2*(2*a^4+6*b^4+-5*a^2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+(1/2)*a^2*b^-1*(a+b*sin(x))^-2*(a^2+-1*b^2)^-1*cos(x)*sin(x)+(1/2)*a^2*b^-2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*(-5*b^2+2*a^2)*cos(x))
@test integrate((a+b*sin(x))^-3*sin(x)^2, x) == :((a^2+-1*b^2)^-5//2*(a^2+2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+(1/2)*a^2*b^-1*(a+b*sin(x))^-2*(a^2+-1*b^2)^-1*cos(x)+-1//2*a*b^-1*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*(a^2+-4*b^2)*cos(x))
@test integrate((a+b*sin(x))^-3*sin(x), x) == :(-1*a*(a+b*sin(x))^-2*(-2*b^2+2*a^2)^-1*cos(x)+-3*a*b*(a^2+-1*b^2)^-5//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1//2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*(a^2+2*b^2)*cos(x))
@test integrate((a+b*sin(x))^-3, x) == :((a^2+-1*b^2)^-5//2*(b^2+2*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+b*(a+b*sin(x))^-2*(-2*b^2+2*a^2)^-1*cos(x)+3//2*a*b*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*cos(x))
@test integrate((a+b*sin(x))^-3*csc(x), x) == :(-1*a^-3*arctanh(cos(x))+-1*b*a^-3*(a^2+-1*b^2)^-5//2*(2*b^4+6*a^4+-5*a^2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1//2*a^-1*b^2*(a+b*sin(x))^-2*(a^2+-1*b^2)^-1*cos(x)+-1//2*a^-2*b^2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*(-2*b^2+5*a^2)*cos(x))
@test integrate((a+b*sin(x))^-3*csc(x)^2, x) == :(3*b*a^-4*arctanh(cos(x))+-1//2*a^-3*(a^2+-1*b^2)^-2*(2*a^4+6*b^4+-11*a^2*b^2)*cot(x)+3*a^-4*b^2*(a^2+-1*b^2)^-5//2*(2*b^4+4*a^4+-5*a^2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1//2*a^-1*b^2*(a+b*sin(x))^-2*(a^2+-1*b^2)^-1*cot(x)+-3//2*a^-2*b^2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*(-1*b^2+2*a^2)*cot(x))
@test integrate((a+b*sin(x))^-3*csc(x)^3, x) == :(-1//2*a^-5*(a^2+12*b^2)*arctanh(cos(x))+-1*a^-5*b^3*(a^2+-1*b^2)^-5//2*(12*b^4+20*a^4+-29*a^2*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x)))+-1//2*a^-3*(a^2+-1*b^2)^-2*(a^4+6*b^4+-10*a^2*b^2)*cot(x)*csc(x)+3//2*b*a^-4*(a^2+-1*b^2)^-2*(2*a^4+4*b^4+-7*a^2*b^2)*cot(x)+-1//2*a^-1*b^2*(a+b*sin(x))^-2*(a^2+-1*b^2)^-1*cot(x)*csc(x)+-1//2*a^-2*b^2*(a+b*sin(x))^-1*(a^2+-1*b^2)^-2*(-4*b^2+7*a^2)*cot(x)*csc(x))
@test integrate((a+b*sin(c+d*x))^-4, x) == :(a*d^-1*(a^2+-1*b^2)^-7//2*(2*a^2+3*b^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*c+(1/2)*d*x)))+b*d^-1*(a+b*sin(c+d*x))^-3*(-3*b^2+3*a^2)^-1*cos(c+d*x)+1//6*b*d^-1*(a+b*sin(c+d*x))^-1*(a^2+-1*b^2)^-3*(4*b^2+11*a^2)*cos(c+d*x)+5//6*a*b*d^-1*(a+b*sin(c+d*x))^-2*(a^2+-1*b^2)^-2*cos(c+d*x))
@test integrate((a+b*sin(e+f*x))^(1/2)*sin(e+f*x), x) == :(-2//3*f^-1*(a+b*sin(e+f*x))^(1/2)*cos(e+f*x)+-1//3*b^-1*f^-1*((a+b)^-1*(a+b*sin(e+f*x)))^(1/2)*(a+b*sin(e+f*x))^-1//2*(-2*b^2+2*a^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*b*(a+b)^-1)+2//3*a*b^-1*f^-1*((a+b)^-1*(a+b*sin(e+f*x)))^-1//2*(a+b*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(e+f*x))^(1/2), x) == :(2*f^-1*((a+b)^-1*(a+b*sin(e+f*x)))^-1//2*(a+b*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(e+f*x))^-1//2*sin(e+f*x), x) == :(2*b^-1*f^-1*((a+b)^-1*(a+b*sin(e+f*x)))^-1//2*(a+b*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*b*(a+b)^-1)+-2*a*b^-1*f^-1*((a+b)^-1*(a+b*sin(e+f*x)))^(1/2)*(a+b*sin(e+f*x))^-1//2*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(e+f*x))^-1//2, x) == :(2*f^-1*((a+b)^-1*(a+b*sin(e+f*x)))^(1/2)*(a+b*sin(e+f*x))^-1//2*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*b*(a+b)^-1))
@test integrate((a+b*sin(c+d*x))^-1//2*sin(c+d*x)^-1//2, x) == :(-2*a^-1*d^-1*(a*(a+b)^-1*(1+-1*csc(c+d*x)))^(1/2)*(a*(a+-1b)^-1*(1+csc(c+d*x)))^(1/2)*(a+b)^(1/2)*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(c+d*x))^(1/2)*sin(c+d*x)^-1//2),-1*(a+-1b)^-1*(a+b))*tan(c+d*x))
@test integrate((c+-1*c*sin(e+f*x))^4*(a+a*sin(e+f*x)), x) == :(7//8*a*x*c^4+1//5*a*f^-1*(c^2+-1*c^2*sin(e+f*x))^2*cos(e+f*x)^3+7//12*a*c^4*f^-1*cos(e+f*x)^3+7//20*a*f^-1*cos(e+f*x)^3*(c^4+-1*c^4*sin(e+f*x))+7//8*a*c^4*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((c+-1*c*sin(e+f*x))^3*(a+a*sin(e+f*x)), x) == :(5//8*a*x*c^3+1//4*a*f^-1*cos(e+f*x)^3*(c^3+-1*c^3*sin(e+f*x))+5//12*a*c^3*f^-1*cos(e+f*x)^3+5//8*a*c^3*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((c+-1*c*sin(e+f*x))^2*(a+a*sin(e+f*x)), x) == :((1/2)*a*x*c^2+1//3*a*c^2*f^-1*cos(e+f*x)^3+(1/2)*a*c^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))*(c+-1*c*sin(e+f*x)), x) == :((1/2)*a*c*x+(1/2)*a*c*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((c+-1*c*sin(e+f*x))^-1*(a+a*sin(e+f*x)), x) == :(-1*a*x*c^-1+2*a*f^-1*(c+-1*c*sin(e+f*x))^-1*cos(e+f*x))
@test integrate((c+-1*c*sin(e+f*x))^-2*(a+a*sin(e+f*x)), x) == :(1//3*a*c*f^-1*(c+-1*c*sin(e+f*x))^-3*cos(e+f*x)^3)
@test integrate((c+-1*c*sin(e+f*x))^-3*(a+a*sin(e+f*x)), x) == :(1//15*a*f^-1*(c+-1*c*sin(e+f*x))^-3*cos(e+f*x)^3+1//5*a*c*f^-1*(c+-1*c*sin(e+f*x))^-4*cos(e+f*x)^3)
@test integrate((c+-1*c*sin(e+f*x))^-4*(a+a*sin(e+f*x)), x) == :(2//35*a*f^-1*(c+-1*c*sin(e+f*x))^-4*cos(e+f*x)^3+1//7*a*c*f^-1*(c+-1*c*sin(e+f*x))^-5*cos(e+f*x)^3+2//105*a*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3*cos(e+f*x)^3)
@test integrate((c+-1*c*sin(e+f*x))^-5*(a+a*sin(e+f*x)), x) == :(1//21*a*f^-1*(c+-1*c*sin(e+f*x))^-5*cos(e+f*x)^3+1//9*a*c*f^-1*(c+-1*c*sin(e+f*x))^-6*cos(e+f*x)^3+2//105*a*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-4*cos(e+f*x)^3+2//315*a*c*f^-1*(c^2+-1*c^2*sin(e+f*x))^-3*cos(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^5, x) == :(9//16*x*a^2*c^5+3//10*a^2*c^5*f^-1*cos(e+f*x)^5+3//14*a^2*f^-1*cos(e+f*x)^5*(c^5+-1*c^5*sin(e+f*x))+1//7*a^2*c^3*f^-1*(c+-1*c*sin(e+f*x))^2*cos(e+f*x)^5+3//8*a^2*c^5*f^-1*cos(e+f*x)^3*sin(e+f*x)+9//16*a^2*c^5*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^4, x) == :(7//16*x*a^2*c^4+1//6*a^2*f^-1*cos(e+f*x)^5*(c^4+-1*c^4*sin(e+f*x))+7//30*a^2*c^4*f^-1*cos(e+f*x)^5+7//16*a^2*c^4*f^-1*cos(e+f*x)*sin(e+f*x)+7//24*a^2*c^4*f^-1*cos(e+f*x)^3*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^3, x) == :(3//8*x*a^2*c^3+1//5*a^2*c^3*f^-1*cos(e+f*x)^5+1//4*a^2*c^3*f^-1*cos(e+f*x)^3*sin(e+f*x)+3//8*a^2*c^3*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^2, x) == :(3//8*x*a^2*c^2+1//4*a^2*c^2*f^-1*cos(e+f*x)^3*sin(e+f*x)+3//8*a^2*c^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x)), x) == :((1/2)*c*x*a^2+-1//3*c*a^2*f^-1*cos(e+f*x)^3+(1/2)*c*a^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-1, x) == :(-3*x*a^2*c^-1+3*a^2*c^-1*f^-1*cos(e+f*x)+2*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-2*cos(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-2, x) == :(x*a^2*c^-2+-2*a^2*f^-1*(c^2+-1*c^2*sin(e+f*x))^-1*cos(e+f*x)+2//3*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-3*cos(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-3, x) == :(1//5*a^2*c^2*f^-1*(c+-1*c*sin(e+f*x))^-5*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-4, x) == :(1//7*a^2*c^2*f^-1*(c+-1*c*sin(e+f*x))^-6*cos(e+f*x)^5+1//35*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-5*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-5, x) == :(2//315*a^2*f^-1*(c+-1*c*sin(e+f*x))^-5*cos(e+f*x)^5+1//9*a^2*c^2*f^-1*(c+-1*c*sin(e+f*x))^-7*cos(e+f*x)^5+2//63*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-6*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-6, x) == :(2//231*a^2*f^-1*(c+-1*c*sin(e+f*x))^-6*cos(e+f*x)^5+1//11*a^2*c^2*f^-1*(c+-1*c*sin(e+f*x))^-8*cos(e+f*x)^5+1//33*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-7*cos(e+f*x)^5+2//1155*a^2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-5*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^6, x) == :(55//128*x*a^3*c^6+1//9*a^3*f^-1*(c^3+-1*c^3*sin(e+f*x))^2*cos(e+f*x)^7+11//56*a^3*c^6*f^-1*cos(e+f*x)^7+11//72*a^3*f^-1*cos(e+f*x)^7*(c^6+-1*c^6*sin(e+f*x))+11//48*a^3*c^6*f^-1*cos(e+f*x)^5*sin(e+f*x)+55//128*a^3*c^6*f^-1*cos(e+f*x)*sin(e+f*x)+55//192*a^3*c^6*f^-1*cos(e+f*x)^3*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^5, x) == :(45//128*x*a^3*c^5+1//8*a^3*f^-1*cos(e+f*x)^7*(c^5+-1*c^5*sin(e+f*x))+9//56*a^3*c^5*f^-1*cos(e+f*x)^7+3//16*a^3*c^5*f^-1*cos(e+f*x)^5*sin(e+f*x)+15//64*a^3*c^5*f^-1*cos(e+f*x)^3*sin(e+f*x)+45//128*a^3*c^5*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^4, x) == :(5//16*x*a^3*c^4+1//7*a^3*c^4*f^-1*cos(e+f*x)^7+1//6*a^3*c^4*f^-1*cos(e+f*x)^5*sin(e+f*x)+5//16*a^3*c^4*f^-1*cos(e+f*x)*sin(e+f*x)+5//24*a^3*c^4*f^-1*cos(e+f*x)^3*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^3, x) == :(5//16*x*a^3*c^3+1//6*a^3*c^3*f^-1*cos(e+f*x)^5*sin(e+f*x)+5//16*a^3*c^3*f^-1*cos(e+f*x)*sin(e+f*x)+5//24*a^3*c^3*f^-1*cos(e+f*x)^3*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^2, x) == :(3//8*x*a^3*c^2+-1//5*a^3*c^2*f^-1*cos(e+f*x)^5+1//4*a^3*c^2*f^-1*cos(e+f*x)^3*sin(e+f*x)+3//8*a^3*c^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x)), x) == :(5//8*c*x*a^3+-5//12*c*a^3*f^-1*cos(e+f*x)^3+-1//4*c*f^-1*cos(e+f*x)^3*(a^3+a^3*sin(e+f*x))+5//8*c*a^3*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-1, x) == :(-15//2*x*a^3*c^-1+5//2*a^3*f^-1*(c+-1*c*sin(e+f*x))^-1*cos(e+f*x)^3+15//2*a^3*c^-1*f^-1*cos(e+f*x)+2*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-3*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-2, x) == :(5*x*a^3*c^-2+-5*a^3*c^-2*f^-1*cos(e+f*x)+-10//3*a^3*f^-1*(c+-1*c*sin(e+f*x))^-2*cos(e+f*x)^3+2//3*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-4*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-3, x) == :(-1*x*a^3*c^-3+2*a^3*f^-1*(c^3+-1*c^3*sin(e+f*x))^-1*cos(e+f*x)+-2//3*a^3*f^-1*(c+-1*c*sin(e+f*x))^-3*cos(e+f*x)^3+2//5*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-5*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-4, x) == :(1//7*a^3*c^3*f^-1*(c+-1*c*sin(e+f*x))^-7*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-5, x) == :(1//9*a^3*c^3*f^-1*(c+-1*c*sin(e+f*x))^-8*cos(e+f*x)^7+1//63*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-7*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-6, x) == :(1//11*a^3*c^3*f^-1*(c+-1*c*sin(e+f*x))^-9*cos(e+f*x)^7+2//99*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-8*cos(e+f*x)^7+2//693*c*a^3*f^-1*(c+-1*c*sin(e+f*x))^-7*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-7, x) == :(2//3003*a^3*f^-1*(c+-1*c*sin(e+f*x))^-7*cos(e+f*x)^7+1//13*a^3*c^3*f^-1*(c+-1*c*sin(e+f*x))^-10*cos(e+f*x)^7+2//429*c*a^3*f^-1*(c+-1*c*sin(e+f*x))^-8*cos(e+f*x)^7+3//143*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-9*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-8, x) == :(8//6435*a^3*f^-1*(c+-1*c*sin(e+f*x))^-8*cos(e+f*x)^7+1//15*a^3*c^3*f^-1*(c+-1*c*sin(e+f*x))^-11*cos(e+f*x)^7+4//195*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-10*cos(e+f*x)^7+4//715*c*a^3*f^-1*(c+-1*c*sin(e+f*x))^-9*cos(e+f*x)^7+8//45045*a^3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-7*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^4, x) == :(-35//2*x*a^-1*c^4+-35//3*a^-1*c^4*f^-1*cos(e+f*x)^3+-14*a*c^4*f^-1*(a+a*sin(e+f*x))^-2*cos(e+f*x)^5+-2*a^3*c^4*f^-1*(a+a*sin(e+f*x))^-4*cos(e+f*x)^7+-35//2*a^-1*c^4*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^3, x) == :(-15//2*x*a^-1*c^3+-15//2*a^-1*c^3*f^-1*cos(e+f*x)+-5//2*c^3*f^-1*(a+a*sin(e+f*x))^-1*cos(e+f*x)^3+-2*a^2*c^3*f^-1*(a+a*sin(e+f*x))^-3*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^2, x) == :(-3*x*a^-1*c^2+-3*a^-1*c^2*f^-1*cos(e+f*x)+-2*a*c^2*f^-1*(a+a*sin(e+f*x))^-2*cos(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x)), x) == :(-1*c*x*a^-1+-2*c*f^-1*(a+a*sin(e+f*x))^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^-1, x) == :(a^-1*c^-1*f^-1*tan(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^-2, x) == :(1//3*a^-1*f^-1*(c^2+-1*c^2*sin(e+f*x))^-1*sec(e+f*x)+2//3*a^-1*c^-2*f^-1*tan(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^-3, x) == :(1//5*a^-1*f^-1*(c^3+-1*c^3*sin(e+f*x))^-1*sec(e+f*x)+2//5*a^-1*c^-3*f^-1*tan(e+f*x)+1//5*a^-1*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-2*sec(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^-4, x) == :(4//35*a^-1*f^-1*(c^2+-1*c^2*sin(e+f*x))^-2*sec(e+f*x)+4//35*a^-1*f^-1*(c^4+-1*c^4*sin(e+f*x))^-1*sec(e+f*x)+8//35*a^-1*c^-4*f^-1*tan(e+f*x)+1//7*a^-1*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3*sec(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^5, x) == :(105//2*x*a^-2*c^5+35*a^-2*c^5*f^-1*cos(e+f*x)^3+42*c^5*f^-1*(a+a*sin(e+f*x))^-2*cos(e+f*x)^5+6*a^2*c^5*f^-1*(a+a*sin(e+f*x))^-4*cos(e+f*x)^7+-2//3*a^4*c^5*f^-1*(a+a*sin(e+f*x))^-6*cos(e+f*x)^9+105//2*a^-2*c^5*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^4, x) == :(35//2*x*a^-2*c^4+35//2*a^-2*c^4*f^-1*cos(e+f*x)+35//6*c^4*f^-1*(a^2+a^2*sin(e+f*x))^-1*cos(e+f*x)^3+-2//3*a^3*c^4*f^-1*(a+a*sin(e+f*x))^-5*cos(e+f*x)^7+14//3*a^4*c^4*f^-1*(a^2+a^2*sin(e+f*x))^-3*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^3, x) == :(5*x*a^-2*c^3+5*a^-2*c^3*f^-1*cos(e+f*x)+10//3*c^3*f^-1*(a+a*sin(e+f*x))^-2*cos(e+f*x)^3+-2//3*a^2*c^3*f^-1*(a+a*sin(e+f*x))^-4*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^2, x) == :(x*a^-2*c^2+2*c^2*f^-1*(a^2+a^2*sin(e+f*x))^-1*cos(e+f*x)+-2//3*a*c^2*f^-1*(a+a*sin(e+f*x))^-3*cos(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x)), x) == :(-1//3*a*c*f^-1*(a+a*sin(e+f*x))^-3*cos(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^-1, x) == :(-1//3*c^-1*f^-1*(a^2+a^2*sin(e+f*x))^-1*sec(e+f*x)+2//3*a^-2*c^-1*f^-1*tan(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^-2, x) == :(a^-2*c^-2*f^-1*tan(e+f*x)+1//3*a^-2*c^-2*f^-1*tan(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^-3, x) == :(1//5*a^-2*f^-1*(c^3+-1*c^3*sin(e+f*x))^-1*sec(e+f*x)^3+4//5*a^-2*c^-3*f^-1*tan(e+f*x)+4//15*a^-2*c^-3*f^-1*tan(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^-4, x) == :(1//7*a^-2*f^-1*(c^2+-1*c^2*sin(e+f*x))^-2*sec(e+f*x)^3+1//7*a^-2*f^-1*(c^4+-1*c^4*sin(e+f*x))^-1*sec(e+f*x)^3+4//7*a^-2*c^-4*f^-1*tan(e+f*x)+4//21*a^-2*c^-4*f^-1*tan(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^-5, x) == :(2//21*a^-2*f^-1*(c^5+-1*c^5*sin(e+f*x))^-1*sec(e+f*x)^3+8//21*a^-2*c^-5*f^-1*tan(e+f*x)+8//63*a^-2*c^-5*f^-1*tan(e+f*x)^3+1//9*a^-2*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-3*sec(e+f*x)^3+2//21*a^-2*c^-3*f^-1*(c+-1*c*sin(e+f*x))^-2*sec(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^5, x) == :(-63//2*x*a^-3*c^5+-63//2*a^-3*c^5*f^-1*cos(e+f*x)+-42//5*c^5*f^-1*(a+a*sin(e+f*x))^-3*cos(e+f*x)^5+-21//2*c^5*f^-1*(a^3+a^3*sin(e+f*x))^-1*cos(e+f*x)^3+-2//5*a^4*c^5*f^-1*(a+a*sin(e+f*x))^-7*cos(e+f*x)^9+6//5*a^2*c^5*f^-1*(a+a*sin(e+f*x))^-5*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^4, x) == :(-7*x*a^-3*c^4+-7*a^-3*c^4*f^-1*cos(e+f*x)+-14//3*a^-1*c^4*f^-1*(a+a*sin(e+f*x))^-2*cos(e+f*x)^3+-2//5*a^3*c^4*f^-1*(a+a*sin(e+f*x))^-6*cos(e+f*x)^7+14//15*a*c^4*f^-1*(a+a*sin(e+f*x))^-4*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^3, x) == :(-1*x*a^-3*c^3+-2*c^3*f^-1*(a^3+a^3*sin(e+f*x))^-1*cos(e+f*x)+2//3*c^3*f^-1*(a+a*sin(e+f*x))^-3*cos(e+f*x)^3+-2//5*a^2*c^3*f^-1*(a+a*sin(e+f*x))^-5*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^2, x) == :(-1//5*a^2*c^2*f^-1*(a+a*sin(e+f*x))^-5*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x)), x) == :(-1//15*c*f^-1*(a+a*sin(e+f*x))^-3*cos(e+f*x)^3+-1//5*a*c*f^-1*(a+a*sin(e+f*x))^-4*cos(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-1, x) == :(-1//5*c^-1*f^-1*(a^3+a^3*sin(e+f*x))^-1*sec(e+f*x)+2//5*a^-3*c^-1*f^-1*tan(e+f*x)+-1//5*a^-1*c^-1*f^-1*(a+a*sin(e+f*x))^-2*sec(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-2, x) == :(-1//5*c^-2*f^-1*(a^3+a^3*sin(e+f*x))^-1*sec(e+f*x)^3+4//5*a^-3*c^-2*f^-1*tan(e+f*x)+4//15*a^-3*c^-2*f^-1*tan(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-3, x) == :(a^-3*c^-3*f^-1*tan(e+f*x)+1//5*a^-3*c^-3*f^-1*tan(e+f*x)^5+2//3*a^-3*c^-3*f^-1*tan(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-4, x) == :(1//7*a^-3*f^-1*(c^4+-1*c^4*sin(e+f*x))^-1*sec(e+f*x)^5+4//7*a^-3*c^-4*f^-1*tan(e+f*x)^3+6//7*a^-3*c^-4*f^-1*tan(e+f*x)+6//35*a^-3*c^-4*f^-1*tan(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-5, x) == :(1//9*a^-3*f^-1*(c^5+-1*c^5*sin(e+f*x))^-1*sec(e+f*x)^5+2//3*a^-3*c^-5*f^-1*tan(e+f*x)+2//15*a^-3*c^-5*f^-1*tan(e+f*x)^5+4//9*a^-3*c^-5*f^-1*tan(e+f*x)^3+1//9*a^-3*c^-3*f^-1*(c+-1*c*sin(e+f*x))^-2*sec(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-6, x) == :(1//11*a^-3*f^-1*(c^2+-1*c^2*sin(e+f*x))^-3*sec(e+f*x)^5+8//99*a^-3*f^-1*(c^3+-1*c^3*sin(e+f*x))^-2*sec(e+f*x)^5+8//99*a^-3*f^-1*(c^6+-1*c^6*sin(e+f*x))^-1*sec(e+f*x)^5+16//33*a^-3*c^-6*f^-1*tan(e+f*x)+16//165*a^-3*c^-6*f^-1*tan(e+f*x)^5+32//99*a^-3*c^-6*f^-1*tan(e+f*x)^3)
@test integrate((c+-1*c*sin(e+f*x))^7//2*(a+a*sin(e+f*x)), x) == :(2//9*a*c^2*f^-1*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)^3+8//21*a*c^3*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)^3+64//105*a*c^4*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)^3+256//315*a*c^5*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^3)
@test integrate((c+-1*c*sin(e+f*x))^5//2*(a+a*sin(e+f*x)), x) == :(2//7*a*c^2*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)^3+16//35*a*c^3*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)^3+64//105*a*c^4*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^3)
@test integrate((c+-1*c*sin(e+f*x))^3//2*(a+a*sin(e+f*x)), x) == :(2//5*a*c^2*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)^3+8//15*a*c^3*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^3)
@test integrate((c+-1*c*sin(e+f*x))^(1/2)*(a+a*sin(e+f*x)), x) == :(2//3*a*c^2*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^3)
@test integrate((c+-1*c*sin(e+f*x))^-1//2*(a+a*sin(e+f*x)), x) == :(-2*a*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+2*a*2^(1/2)*c^-1//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((c+-1*c*sin(e+f*x))^-3//2*(a+a*sin(e+f*x)), x) == :(a*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-1//2*a*2^(1/2)*c^-3//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((c+-1*c*sin(e+f*x))^-5//2*(a+a*sin(e+f*x)), x) == :((1/2)*a*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+-1//8*a*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-1//16*a*2^(1/2)*c^-5//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((c+-1*c*sin(e+f*x))^-7//2*(a+a*sin(e+f*x)), x) == :(1//3*a*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)+-1//24*a*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+-1//32*a*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-1//64*a*2^(1/2)*c^-7//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^7//2, x) == :(2//11*a^2*c^3*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)^5+8//33*a^2*c^4*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)^5+64//231*a^2*c^5*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^5+256//1155*a^2*c^6*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^5//2, x) == :(2//9*a^2*c^3*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)^5+16//63*a^2*c^4*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^5+64//315*a^2*c^5*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^3//2, x) == :(2//7*a^2*c^3*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^5+8//35*a^2*c^4*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^(1/2), x) == :(2//5*a^2*c^3*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-4*a^2*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+4*2^(1/2)*a^2*c^-1//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))+-2//3*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-3//2, x) == :(c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^3+-3*2^(1/2)*a^2*c^-3//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))+3*a^2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-5//2, x) == :((1/2)*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)^3+-3//4*a^2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+3//8*2^(1/2)*a^2*c^-5//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-7//2, x) == :(-1//4*a^2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+1//3*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x)^3+1//16*a^2*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+1//32*2^(1/2)*a^2*c^-7//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^2*(c+-1*c*sin(e+f*x))^-9//2, x) == :(-1//8*a^2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)+1//4*c*a^2*f^-1*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)^3+1//64*a^2*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+3//256*a^2*c^-3*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+3//512*2^(1/2)*a^2*c^-9//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^7//2, x) == :(2//13*a^3*c^4*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)^7+24//143*a^3*c^5*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^7+64//429*a^3*c^6*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^7+256//3003*a^3*c^7*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^5//2, x) == :(2//11*a^3*c^4*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^7+16//99*a^3*c^5*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^7+64//693*a^3*c^6*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^3//2, x) == :(2//9*a^3*c^4*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^7+8//63*a^3*c^5*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^(1/2), x) == :(2//7*a^3*c^4*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)^7)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-8*a^3*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+8*2^(1/2)*a^3*c^-1//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))+-4//3*c*a^3*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^3+-2//5*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-3//2, x) == :(5//3*a^3*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)^3+a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)^5+-10*2^(1/2)*a^3*c^-3//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))+10*a^3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-5//2, x) == :(-5//4*a^3*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)^3+(1/2)*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x)^5+-15//4*a^3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+15//4*2^(1/2)*a^3*c^-5//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-7//2, x) == :(-5//12*a^3*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)^3+-5//16*2^(1/2)*a^3*c^-7//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))+1//3*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)^5+5//8*a^3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-9//2, x) == :(-5//24*a^3*f^-1*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x)^3+-5//128*a^3*c^-3*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-5//256*2^(1/2)*a^3*c^-9//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))+1//4*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-13//2*cos(e+f*x)^5+5//32*a^3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+-1*c*sin(e+f*x))^-11//2, x) == :(-1//8*a^3*f^-1*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)^3+-3//512*a^3*c^-4*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-3//1024*2^(1/2)*a^3*c^-11//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))+-1//128*a^3*c^-3*f^-1*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+1//5*a^3*c^2*f^-1*(c+-1*c*sin(e+f*x))^-15//2*cos(e+f*x)^5+1//16*a^3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^7//2, x) == :(2//5*a^-1*f^-1*(c+-1*c*sin(e+f*x))^7//2*sec(e+f*x)+-256//5*a^-1*c^3*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x)+8//5*c*a^-1*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)+64//5*a^-1*c^2*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^5//2, x) == :(2//3*a^-1*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)+-64//3*a^-1*c^2*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x)+16//3*c*a^-1*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^3//2, x) == :(2*a^-1*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)+-8*c*a^-1*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^(1/2), x) == :(-2*a^-1*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^-1//2, x) == :((1/2)*2^(1/2)*a^-1*c^-1//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))+-1*a^-1*c^-1*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^-3//2, x) == :(3//4*a^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-1*a^-1*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-1//2*sec(e+f*x)+3//8*2^(1/2)*a^-1*c^-3//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1*(c+-1*c*sin(e+f*x))^-5//2, x) == :(-5//8*a^-1*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-1//2*sec(e+f*x)+1//4*a^-1*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*sec(e+f*x)+15//32*a^-1*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+15//64*2^(1/2)*a^-1*c^-5//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^9//2, x) == :(32//15*a^-2*f^-1*(c+-1*c*sin(e+f*x))^9//2*sec(e+f*x)^3+-1024//5*a^-2*c^2*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^3+2//5*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^11//2*sec(e+f*x)^3+128//5*c*a^-2*f^-1*(c+-1*c*sin(e+f*x))^7//2*sec(e+f*x)^3+4096//15*a^-2*c^3*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^7//2, x) == :(8*a^-2*f^-1*(c+-1*c*sin(e+f*x))^7//2*sec(e+f*x)^3+-64*c*a^-2*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^3+2//3*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^9//2*sec(e+f*x)^3+256//3*a^-2*c^2*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^5//2, x) == :(-16*a^-2*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^3+2*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^7//2*sec(e+f*x)^3+64//3*c*a^-2*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^3//2, x) == :(8//3*a^-2*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)^3+-2*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^(1/2), x) == :(-2//3*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)^3)
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-1//2*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x)+-1//3*a^-2*c^-2*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)^3+1//4*2^(1/2)*a^-2*c^-1//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^-3//2, x) == :(5//8*a^-2*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-5//6*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-1//2*sec(e+f*x)+-1//3*a^-2*c^-2*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x)^3+5//16*2^(1/2)*a^-2*c^-3//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-2*(c+-1*c*sin(e+f*x))^-5//2, x) == :(-35//48*a^-2*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-1//2*sec(e+f*x)+-1//3*a^-2*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-1//2*sec(e+f*x)^3+7//24*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*sec(e+f*x)+35//64*a^-2*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+35//128*2^(1/2)*a^-2*c^-5//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^9//2, x) == :(-128*a^-3*f^-1*(c+-1*c*sin(e+f*x))^9//2*sec(e+f*x)^5+-4096//15*a^-3*c^2*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^5+2//3*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^13//2*sec(e+f*x)^5+32//3*a^-3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^11//2*sec(e+f*x)^5+1024//3*c*a^-3*f^-1*(c+-1*c*sin(e+f*x))^7//2*sec(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^7//2, x) == :(64*a^-3*f^-1*(c+-1*c*sin(e+f*x))^7//2*sec(e+f*x)^5+-24*a^-3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^9//2*sec(e+f*x)^5+2*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^11//2*sec(e+f*x)^5+-256//5*c*a^-3*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^5//2, x) == :(-64//15*a^-3*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^5+-2*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^9//2*sec(e+f*x)^5+16//3*a^-3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^7//2*sec(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^3//2, x) == :(-2//3*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^7//2*sec(e+f*x)^5+8//15*a^-3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^(1/2), x) == :(-2//5*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^5)
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-1//4*a^-3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x)+-1//5*a^-3*c^-3*f^-1*(c+-1*c*sin(e+f*x))^5//2*sec(e+f*x)^5+-1//6*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)^3+1//8*2^(1/2)*a^-3*c^-1//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-3//2, x) == :(7//16*a^-3*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-7//12*a^-3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-1//2*sec(e+f*x)+-7//30*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x)^3+-1//5*a^-3*c^-3*f^-1*(c+-1*c*sin(e+f*x))^3//2*sec(e+f*x)^5+7//32*2^(1/2)*a^-3*c^-3//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3*(c+-1*c*sin(e+f*x))^-5//2, x) == :(-21//32*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-1//2*sec(e+f*x)+-3//10*a^-3*c^-2*f^-1*(c+-1*c*sin(e+f*x))^-1//2*sec(e+f*x)^3+-1//5*a^-3*c^-3*f^-1*(c+-1*c*sin(e+f*x))^(1/2)*sec(e+f*x)^5+21//80*a^-3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*sec(e+f*x)+63//128*a^-3*c^-1*f^-1*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+63//256*2^(1/2)*a^-3*c^-5//2*f^-1*arctanh((1/2)*2^(1/2)*c^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^7//2, x) == :(-1//4*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^5//2, x) == :(-1//3*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^3//2, x) == :(-1//2*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^(1/2), x) == :(-1*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-1*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-3//2, x) == :(a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-5//2, x) == :((1/2)*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-7//2, x) == :(1//3*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^7//2, x) == :(-1//5*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x)+-1//10*a^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^5//2, x) == :(-1//4*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x)+-1//6*a^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^3//2, x) == :(-1//3*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+-1//3*a^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^(1/2), x) == :((1/2)*c*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-1*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+-2*a^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-3//2, x) == :(a*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+a^2*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-5//2, x) == :(1//4*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-7//2, x) == :(1//6*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)+1//24*c^-1*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-9//2, x) == :(1//4*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x)+-1//12*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-11//2, x) == :(1//5*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)+-1//20*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^7//2, x) == :(-2//15*a^2*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x)+-1//6*a*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x)+-1//15*a^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^5//2, x) == :(-2//15*a^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x)+-1//5*a*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x)+-1//5*a^2*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^3//2, x) == :(1//4*c*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+1//6*c^2*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^(1/2), x) == :(1//3*c*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-2*a^2*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+-1//2*a*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+-4*a^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-3//2, x) == :(a*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+2*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+4*a^3*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-5//2, x) == :((1/2)*a*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+-1*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-1*a^3*c^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-7//2, x) == :(1//6*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-9//2, x) == :(1//8*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x)+1//48*c^-1*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-11//2, x) == :(1//10*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)+1//40*c^-1*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x)+1//240*c^-2*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-13//2, x) == :(1//6*a*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-13//2*cos(e+f*x)+-1//15*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)+1//60*a^3*c^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^9//2, x) == :(-3//28*a^2*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^9//2*cos(e+f*x)+-1//8*a*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^9//2*cos(e+f*x)+-1//14*a^3*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^9//2*cos(e+f*x)+-1//35*a^4*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^9//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^7//2, x) == :(-4//35*a^3*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x)+-2//35*a^4*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x)+-1//7*a*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x)+-1//7*a^2*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^5//2, x) == :(1//6*c*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+1//15*c^3*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+2//15*c^2*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^3//2, x) == :(1//5*c*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+1//10*c^2*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^(1/2), x) == :(1//4*c*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-1*a^2*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+-4*a^3*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+-1//3*a*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+-8*a^4*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-3//2, x) == :(a*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+6*a^3*c^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+3//2*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+12*a^4*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-5//2, x) == :((1/2)*a*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+-3*a^3*c^-2*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+-3//2*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-6*a^4*c^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-7//2, x) == :(1//3*a*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-7//2*cos(e+f*x)+a^3*c^-2*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-1//2*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+a^4*c^-3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+-1*sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-9//2, x) == :(1//8*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-11//2, x) == :(1//10*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)+1//80*c^-1*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-13//2, x) == :(1//12*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-13//2*cos(e+f*x)+1//60*c^-1*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)+1//480*c^-2*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-15//2, x) == :(1//14*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-15//2*cos(e+f*x)+1//56*c^-1*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-13//2*cos(e+f*x)+1//280*c^-2*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)+1//2240*c^-3*f^-1*(a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-9//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^7//2*(c+-1*c*sin(e+f*x))^-17//2, x) == :(1//8*a*f^-1*(a+a*sin(e+f*x))^5//2*(c+-1*c*sin(e+f*x))^-17//2*cos(e+f*x)+-3//56*a^2*c^-1*f^-1*(a+a*sin(e+f*x))^3//2*(c+-1*c*sin(e+f*x))^-15//2*cos(e+f*x)+-1//280*a^4*c^-3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-11//2*cos(e+f*x)+1//56*a^3*c^-2*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+-1*c*sin(e+f*x))^-13//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^5//2, x) == :((1/2)*c*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+2*c^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+4*c^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^3//2, x) == :(c*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+2*c^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^(1/2), x) == :(c*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2, x) == :(f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-3//2, x) == :((1/2)*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+(1/2)*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-5//2, x) == :(1//4*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+1//4*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+1//4*c^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^7//2, x) == :(-1*c*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x)+-6*a^-1*c^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+-3//2*a^-1*c^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+-12*a^-1*c^4*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^5//2, x) == :(-1*c*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+-2*a^-1*c^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+-4*a^-1*c^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^3//2, x) == :(-1*c*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+-1*a^-1*c^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^(1/2), x) == :(-1*c*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+(1/2)*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-3//2, x) == :(-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+(1/2)*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+(1/2)*a^-1*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-5//2, x) == :(-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+3//8*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+3//8*a^-1*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+3//8*a^-1*c^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^9//2, x) == :(-1//2*c*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^7//2*cos(e+f*x)+2*a^-1*c^2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x)+3*a^-2*c^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+12*a^-2*c^4*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+24*a^-2*c^5*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^7//2, x) == :(-1//2*c*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^5//2*cos(e+f*x)+3*a^-2*c^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+3//2*a^-1*c^2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+6*a^-2*c^4*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^5//2, x) == :(-1//2*c*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+a^-1*c^2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+a^-2*c^3*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)*log(1+sin(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^3//2, x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^(1/2), x) == :(-1//2*c*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^-1//2, x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+-1//4*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x)+1//4*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^-3//2, x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+-3//8*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+3//8*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+3//8*a^-2*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^-5//2, x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+-1//2*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+3//8*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-5//2*cos(e+f*x)+3//8*a^-2*c^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-3//2*cos(e+f*x)+3//8*a^-2*c^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-1*c*sin(e+f*x))^-1//2*arctanh(sin(e+f*x))*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^5//2, x) == :(2*c*f^-1*(5+2m)^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^3//2*cos(e+f*x)+16*c^2*f^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^(1/2)*(15+4*m^2+16m)^-1*cos(e+f*x)+64*c^3*f^-1*(5+2m)^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^-1//2*(3+4*m^2+8m)^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^3//2, x) == :(2*c*f^-1*(3+2m)^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^(1/2)*cos(e+f*x)+8*c^2*f^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^-1//2*(3+4*m^2+8m)^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^(1/2), x) == :(2*c*f^-1*(1+2m)^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^(-2+-1m), x) == :(f^-1*(3+2m)^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^(-2+-1m)*cos(e+f*x)+c^-1*f^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^(-1+-1m)*(3+4*m^2+8m)^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^(-1+-1m), x) == :(f^-1*(1+2m)^-1*(a+a*sin(e+f*x))^m*(c+-1*c*sin(e+f*x))^(-1+-1m)*cos(e+f*x))
@test integrate((c+d*sin(e+f*x))^4*(a+a*sin(e+f*x)), x) == :(1//8*a*x*(3*d^4+8*c^4+12*c*d^3+16*d*c^3+24*c^2*d^2)+-1//5*a*f^-1*(c+d*sin(e+f*x))^4*cos(e+f*x)+-1//30*a*f^-1*(12*c^4+16*d^4+80*c*d^3+95*d*c^3+112*c^2*d^2)*cos(e+f*x)+-1//20*a*f^-1*(c+d*sin(e+f*x))^3*(4c+5d)*cos(e+f*x)+-1//60*a*f^-1*(c+d*sin(e+f*x))^2*(12*c^2+16*d^2+35*c*d)*cos(e+f*x)+-1//120*a*d*f^-1*(24*c^3+45*d^3+116*c*d^2+130*d*c^2)*cos(e+f*x)*sin(e+f*x))
@test integrate((c+d*sin(e+f*x))^3*(a+a*sin(e+f*x)), x) == :(1//8*a*x*(3*d^3+8*c^3+12*c*d^2+12*d*c^2)+-1//4*a*f^-1*(c+d*sin(e+f*x))^3*cos(e+f*x)+-1//6*a*f^-1*(3*c^3+4*d^3+12*c*d^2+16*d*c^2)*cos(e+f*x)+-1//12*a*f^-1*(c+d*sin(e+f*x))^2*(3c+4d)*cos(e+f*x)+-1//24*a*d*f^-1*(6*c^2+9*d^2+20*c*d)*cos(e+f*x)*sin(e+f*x))
@test integrate((c+d*sin(e+f*x))^2*(a+a*sin(e+f*x)), x) == :((1/2)*a*x*(d^2+2*c^2+2*c*d)+-2//3*a*f^-1*(c^2+d^2+3*c*d)*cos(e+f*x)+-1//3*a*f^-1*(c+d*sin(e+f*x))^2*cos(e+f*x)+-1//6*a*d*f^-1*(2c+3d)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))*(c+d*sin(e+f*x)), x) == :((1/2)*a*x*(d+2c)+-1*a*f^-1*(c+d)*cos(e+f*x)+-1//2*a*d*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate(a+a*sin(e+f*x), x) == :(a*x+-1*a*f^-1*cos(e+f*x))
@test integrate((c+d*sin(e+f*x))^-1*(a+a*sin(e+f*x)), x) == :(a*x*d^-1+-2*a*d^-1*f^-1*(c^2+-1*d^2)^-1//2*(c+-1d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((c+d*sin(e+f*x))^-2*(a+a*sin(e+f*x)), x) == :(-1*a*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-1*cos(e+f*x)+2*a*f^-1*(c+d)^-1*(c^2+-1*d^2)^-1//2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((c+d*sin(e+f*x))^-3*(a+a*sin(e+f*x)), x) == :(-1*a*f^-1*(c+d*sin(e+f*x))^-2*(2c+2d)^-1*cos(e+f*x)+a*f^-1*(c+d)^-1*(c^2+-1*d^2)^-3//2*(-1d+2c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1*a*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-1*(-2d+2c)^-1*(c+-2d)*cos(e+f*x))
@test integrate((c+d*sin(e+f*x))^-4*(a+a*sin(e+f*x)), x) == :(-1*a*f^-1*(c+d*sin(e+f*x))^-3*(3c+3d)^-1*cos(e+f*x)+a*f^-1*(c+d)^-1*(c^2+-1*d^2)^-5//2*(d^2+2*c^2+-2*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1*a*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-2*(-6d+6c)^-1*(-3d+2c)*cos(e+f*x)+-1//6*a*f^-1*(c+d)^-3*(c+-1d)^-2*(c+d*sin(e+f*x))^-1*(c+-4d)*(-1d+2c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^4, x) == :(1//16*x*a^2*(11*d^4+24*c^4+48*c*d^3+64*d*c^3+84*c^2*d^2)+-1//6*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^5*cos(e+f*x)+1//60*a^2*d^-1*f^-1*(-64*d^5+4*c^5+-448*c^2*d^3+-311*c^3*d^2+-288*c*d^4+-48*d*c^4)*cos(e+f*x)+1//240*a^2*f^-1*(-165*d^4+8*c^4+-464*c*d^3+-438*c^2*d^2+-96*d*c^3)*cos(e+f*x)*sin(e+f*x)+1//30*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^4*(c+-12d)*cos(e+f*x)+1//120*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^2*(-64*d^3+4*c^3+-123*c*d^2+-48*d*c^2)*cos(e+f*x)+1//120*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^3*(-55*d^2+4*c^2+-48*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^3, x) == :(3//8*x*a^2*(d+2c)*(2*c^2+2*d^2+3*c*d)+-1//5*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^4*cos(e+f*x)+1//10*a^2*d^-1*f^-1*(c^4+-12*d^4+-44*c^2*d^2+-40*c*d^3+-10*d*c^3)*cos(e+f*x)+1//40*a^2*f^-1*(-30*d^3+2*c^3+-57*c*d^2+-20*d*c^2)*cos(e+f*x)*sin(e+f*x)+1//20*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^2*(c^2+-12*d^2+-10*c*d)*cos(e+f*x)+1//20*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^3*(c+-10d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^2, x) == :(1//8*x*a^2*(7*d^2+12*c^2+16*c*d)+-1//6*a^2*f^-1*(7*d^2+12*c^2+16*c*d)*cos(e+f*x)+-1//4*a^-1*d^2*f^-1*(a+a*sin(e+f*x))^3*cos(e+f*x)+-1//12*d*f^-1*(a+a*sin(e+f*x))^2*(-1d+8c)*cos(e+f*x)+-1//24*a^2*f^-1*(7*d^2+12*c^2+16*c*d)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x)), x) == :((1/2)*x*a^2*(2d+3c)+-2//3*a^2*f^-1*(2d+3c)*cos(e+f*x)+-1//3*d*f^-1*(a+a*sin(e+f*x))^2*cos(e+f*x)+-1//6*a^2*f^-1*(2d+3c)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^2, x) == :(3//2*x*a^2+-2*a^2*f^-1*cos(e+f*x)+-1//2*a^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^-1, x) == :(-1*x*a^2*d^-2*(c+-2d)+-1*a^2*d^-1*f^-1*cos(e+f*x)+2*a^2*d^-2*f^-1*(c+-1d)^2*(c^2+-1*d^2)^-1//2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^-3, x) == :(3*a^2*f^-1*(c+d)^-2*(c^2+-1*d^2)^-1//2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+(1/2)*a^2*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-2*(c+-1d)*cos(e+f*x)+-1//2*a^2*d^-1*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-1*(c+4d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^-4, x) == :(a^2*f^-1*(c+d)^-3*(c+-1d)^-1*(c^2+-1*d^2)^-1//2*(-2d+3c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//6*a^2*d^-1*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-2*(c+6d)*cos(e+f*x)+1//3*a^2*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-3*(c+-1d)*cos(e+f*x)+-1*a^2*d^-1*f^-1*(c+d)^-3*(c+d*sin(e+f*x))^-1*(-6d+6c)^-1*(c^2+-10*d^2+6*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^-5, x) == :(-1//12*a^2*d^-1*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-3*(c+8d)*cos(e+f*x)+1//4*a^2*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-4*(c+-1d)*cos(e+f*x)+1//4*a^2*f^-1*(c+d)^-4*(c+-1d)^-2*(c^2+-1*d^2)^-1//2*(7*d^2+12*c^2+-16*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1*a^2*d^-1*f^-1*(c+d)^-3*(c+d*sin(e+f*x))^-2*(-24d+24c)^-1*(-21*d^2+2*c^2+16*c*d)*cos(e+f*x)+-1//24*a^2*d^-1*f^-1*(c+d)^-4*(c+-1d)^-2*(c+d*sin(e+f*x))^-1*(2*c^3+32*d^3+-59*c*d^2+16*d*c^2)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3, x) == :(5//2*x*a^3+-4*a^3*f^-1*cos(e+f*x)+1//3*a^3*f^-1*cos(e+f*x)^3+-3//2*a^3*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-1, x) == :((1/2)*x*a^3*d^-3*(2*c^2+7*d^2+-6*c*d)+-1//2*d^-1*f^-1*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+(1/2)*a^3*d^-2*f^-1*(-5d+2c)*cos(e+f*x)+-2*a^3*d^-3*f^-1*(c+-1d)^3*(c^2+-1*d^2)^-1//2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-2, x) == :(-1*x*a^3*d^-3*(-3d+2c)+-2*c*a^3*d^-2*f^-1*(c+d)^-1*cos(e+f*x)+d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-1*(c+-1d)*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+2*a^3*d^-3*f^-1*(c+d)^-1*(c+-1d)^2*(c^2+-1*d^2)^-1//2*(2c+3d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-3, x) == :(x*a^3*d^-3+(1/2)*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-2*(c+-1d)*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+(1/2)*a^3*d^-2*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-1*(c+-1d)*(2c+5d)*cos(e+f*x)+-1*a^3*d^-3*f^-1*(c+d)^-2*(c^2+-1*d^2)^-1//2*(c+-1d)*(2*c^2+7*d^2+6*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-4, x) == :(5*a^3*f^-1*(c+d)^-3*(c^2+-1*d^2)^-1//2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//6*a^3*d^-2*f^-1*(c+d)^-3*(c+d*sin(e+f*x))^-1*(2*c^2+22*d^2+9*c*d)*cos(e+f*x)+1//3*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-3*(c+-1d)*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+1//6*a^3*d^-2*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-2*(c+-1d)*(2c+7d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-5, x) == :(5*a^3*f^-1*(c+d)^-4*(c^2+-1*d^2)^-1//2*(-4d+4c)^-1*(-3d+4c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//24*a^3*d^-2*f^-1*(c+d)^-3*(c+d*sin(e+f*x))^-2*(2*c^2+45*d^2+12*c*d)*cos(e+f*x)+1//4*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-4*(c+-1d)*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+-1*a^3*d^-2*f^-1*(c+d)^-4*(c+d*sin(e+f*x))^-1*(-24d+24c)^-1*(-72*d^3+2*c^3+12*d*c^2+43*c*d^2)*cos(e+f*x)+1//12*a^3*d^-2*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-3*(c+-1d)*(2c+9d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^4, x) == :((1/2)*d*x*a^-1*(-3*d^3+8*c^3+-12*d*c^2+12*c*d^2)+-1*f^-1*(a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^3*(c+-1d)*cos(e+f*x)+2//3*d*a^-1*f^-1*(-4*d^3+3*c^3+-16*d*c^2+12*c*d^2)*cos(e+f*x)+1//3*d*a^-1*f^-1*(c+d*sin(e+f*x))^2*(-4d+3c)*cos(e+f*x)+1//6*a^-1*d^2*f^-1*(6*c^2+9*d^2+-20*c*d)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^3, x) == :(3//2*d*x*a^-1*(d^2+2*c^2+-2*c*d)+-1*f^-1*(a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^2*(c+-1d)*cos(e+f*x)+2*d*a^-1*f^-1*(c^2+d^2+-3*c*d)*cos(e+f*x)+(1/2)*a^-1*d^2*f^-1*(-3d+2c)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^2, x) == :(d*x*a^-1*(-1d+2c)+-1*a^-1*d^2*f^-1*cos(e+f*x)+-1*a^-1*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x)), x) == :(d*x*a^-1+-1*f^-1*(a+a*sin(e+f*x))^-1*(c+-1d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-1, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*(c+-1d)^-1*cos(e+f*x)+-2*d*a^-1*f^-1*(c+-1d)^-1*(c^2+-1*d^2)^-1//2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*(c+-1d)^-1*(c+d*sin(e+f*x))^-1*cos(e+f*x)+-2*d*a^-1*f^-1*(c+-1d)^-1*(c^2+-1*d^2)^-3//2*(d+2c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1*d*a^-1*f^-1*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-1*(c+2d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-3, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*(c+-1d)^-1*(c+d*sin(e+f*x))^-2*cos(e+f*x)+-3*d*a^-1*f^-1*(c+-1d)^-1*(c^2+-1*d^2)^-5//2*(d^2+2*c^2+2*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//2*d*a^-1*f^-1*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-2*(2c+3d)*cos(e+f*x)+-1//2*d*a^-1*f^-1*(c+d)^-2*(c+-1d)^-3*(c+d*sin(e+f*x))^-1*(c+4d)*(d+2c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^5, x) == :((1/2)*x*a^-2*d^2*(-5d+10c)*(2*c^2+2*d^2+-3*c*d)+-1//3*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^4*(c+-1d)*cos(e+f*x)+2//3*d*a^-2*f^-1*(c^4+-12*d^4+-44*c^2*d^2+10*d*c^3+40*c*d^3)*cos(e+f*x)+1//3*d*a^-2*f^-1*(c+d*sin(e+f*x))^2*(c^2+-12*d^2+10*c*d)*cos(e+f*x)+1//6*a^-2*d^2*f^-1*(2*c^3+30*d^3+-57*c*d^2+20*d*c^2)*cos(e+f*x)*sin(e+f*x)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+d*sin(e+f*x))^3*(c+-1d)*(c+10d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^4, x) == :((1/2)*x*a^-2*d^2*(7*d^2+12*c^2+-16*c*d)+-1//3*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^3*(c+-1d)*cos(e+f*x)+2//3*d*a^-2*f^-1*(c^3+8*d^3+-20*c*d^2+8*d*c^2)*cos(e+f*x)+1//6*a^-2*d^2*f^-1*(-21*d^2+2*c^2+16*c*d)*cos(e+f*x)*sin(e+f*x)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+d*sin(e+f*x))^2*(c+-1d)*(c+8d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^3, x) == :(x*a^-2*d^2*(-2d+3c)+-1//3*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^2*(c+-1d)*cos(e+f*x)+1//3*a^-2*d^2*f^-1*(c+-4d)*cos(e+f*x)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^2*(c+6d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^2, x) == :(x*a^-2*d^2+-1//3*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)*(c+d*sin(e+f*x))*cos(e+f*x)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)*(c+4d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x)), x) == :(-1//3*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)*cos(e+f*x)+-1//3*f^-1*(a^2+a^2*sin(e+f*x))^-1*(c+2d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2, x) == :(-1//3*f^-1*(a+a*sin(e+f*x))^-2*cos(e+f*x)+-1//3*f^-1*(a^2+a^2*sin(e+f*x))^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-1, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-2*(-3d+3c)^-1*cos(e+f*x)+2*a^-2*d^2*f^-1*(c+-1d)^-2*(c^2+-1*d^2)^-1//2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^-2*(c+-4d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-1*(-3d+3c)^-1*cos(e+f*x)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-1*(c+-6d)*cos(e+f*x)+2*a^-2*d^2*f^-1*(c+d)^-1*(c+-1d)^-3*(c^2+-1*d^2)^-1//2*(2d+3c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//3*d*a^-2*f^-1*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-1*(c^2+-10*d^2+-6*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-3, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-2*(-3d+3c)^-1*cos(e+f*x)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-2*(c+-8d)*cos(e+f*x)+a^-2*d^2*f^-1*(c+d)^-2*(c+-1d)^-4*(c^2+-1*d^2)^-1//2*(7*d^2+12*c^2+16*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//6*d*a^-2*f^-1*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-2*(-21*d^2+2*c^2+-16*c*d)*cos(e+f*x)+-1//6*d*a^-2*f^-1*(c+d)^-2*(c+-1d)^-4*(c+d*sin(e+f*x))^-1*(-32*d^3+2*c^3+-59*c*d^2+-16*d*c^2)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^6, x) == :((1/2)*x*a^-3*d^3*(-23*d^3+40*c^3+-90*d*c^2+78*c*d^2)+-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^5*(c+-1d)*cos(e+f*x)+2//15*d*a^-3*f^-1*(-136*d^5+2*c^5+-472*c^2*d^3+18*d*c^4+107*c^3*d^2+456*c*d^4)*cos(e+f*x)+-1//15*f^-1*(c+d*sin(e+f*x))^3*(a^3+a^3*sin(e+f*x))^-1*(c+-1d)*(2*c^2+115*d^2+18*c*d)*cos(e+f*x)+1//15*d*a^-3*f^-1*(c+d*sin(e+f*x))^2*(-136*d^3+2*c^3+18*d*c^2+111*c*d^2)*cos(e+f*x)+1//30*a^-3*d^2*f^-1*(4*c^4+345*d^4+-626*c*d^3+36*d*c^3+216*c^2*d^2)*cos(e+f*x)*sin(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^4*(c+-1d)*(2c+13d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^5, x) == :((1/2)*x*a^-3*d^3*(13*d^2+20*c^2+-30*c*d)+-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^4*(c+-1d)*cos(e+f*x)+2//15*d*a^-3*f^-1*(2*c^4+76*d^4+-180*c*d^3+15*d*c^3+72*c^2*d^2)*cos(e+f*x)+-1//15*f^-1*(c+d*sin(e+f*x))^2*(a^3+a^3*sin(e+f*x))^-1*(c+-1d)*(2*c^2+76*d^2+15*c*d)*cos(e+f*x)+1//30*a^-3*d^2*f^-1*(-195*d^3+4*c^3+30*d*c^2+146*c*d^2)*cos(e+f*x)*sin(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^3*(c+-1d)*(2c+11d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^4, x) == :(x*a^-3*d^3*(-3d+4c)+-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^3*(c+-1d)*cos(e+f*x)+-1//15*f^-1*(c+-1d)^2*(a^3+a^3*sin(e+f*x))^-1*(2*c^2+45*d^2+12*c*d)*cos(e+f*x)+1//15*a^-3*d^2*f^-1*(-27*d^2+2*c^2+10*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^2*(c+-1d)*(2c+9d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^3, x) == :(x*a^-3*d^3+-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^2*(c+-1d)*cos(e+f*x)+-1//15*f^-1*(a^3+a^3*sin(e+f*x))^-1*(c+-1d)*(2*c^2+29*d^2+11*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)^2*(2c+7d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^2, x) == :(-1//15*f^-1*(a^3+a^3*sin(e+f*x))^-1*(2*c^2+7*d^2+6*c*d)*cos(e+f*x)+-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+-1d)*(c+d*sin(e+f*x))*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)*(2c+5d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x)), x) == :(-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+-1d)*cos(e+f*x)+-1//15*f^-1*(a^3+a^3*sin(e+f*x))^-1*(2c+3d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(2c+3d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3, x) == :(-2//15*f^-1*(a^3+a^3*sin(e+f*x))^-1*cos(e+f*x)+-1//5*f^-1*(a+a*sin(e+f*x))^-3*cos(e+f*x)+-2//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-1, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3*(-5d+5c)^-1*cos(e+f*x)+-1//15*f^-1*(c+-1d)^-3*(a^3+a^3*sin(e+f*x))^-1*(2*c^2+22*d^2+-9*c*d)*cos(e+f*x)+-2*a^-3*d^3*f^-1*(c+-1d)^-3*(c^2+-1*d^2)^-1//2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)^-2*(-7d+2c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-1*(-5d+5c)^-1*cos(e+f*x)+-1//15*f^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-1*(a^3+a^3*sin(e+f*x))^-1*(2*c^2+45*d^2+-12*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)^-2*(c+d*sin(e+f*x))^-1*(-9d+2c)*cos(e+f*x)+-2*a^-3*d^3*f^-1*(c+d)^-1*(c+-1d)^-4*(c^2+-1*d^2)^-1//2*(3d+4c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//15*d*a^-3*f^-1*(c+d)^-1*(c+-1d)^-4*(c+d*sin(e+f*x))^-1*(2*c^3+72*d^3+-12*d*c^2+43*c*d^2)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-3, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-2*(-5d+5c)^-1*cos(e+f*x)+-1//15*f^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-2*(a^3+a^3*sin(e+f*x))^-1*(2*c^2+76*d^2+-15*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)^-2*(c+d*sin(e+f*x))^-2*(-11d+2c)*cos(e+f*x)+-1*a^-3*d^3*f^-1*(c+d)^-2*(c+-1d)^-5*(c^2+-1*d^2)^-1//2*(13*d^2+20*c^2+30*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//30*d*a^-3*f^-1*(c+d)^-1*(c+-1d)^-4*(c+d*sin(e+f*x))^-2*(4*c^3+195*d^3+-30*d*c^2+146*c*d^2)*cos(e+f*x)+-1//30*d*a^-3*f^-1*(c+d)^-2*(c+-1d)^-5*(c+d*sin(e+f*x))^-1*(4*c^4+304*d^4+-30*d*c^3+142*c^2*d^2+525*c*d^3)*cos(e+f*x))
@test integrate((1+sin(x))^-4*(A+B*sin(x)), x) == :(-1*(105+105*sin(x))^-1*(6A+8B)*cos(x)+-1//7*(1+sin(x))^-4*(A+-1B)*cos(x)+-1//35*(1+sin(x))^-3*(3A+4B)*cos(x)+-1//105*(1+sin(x))^-2*(6A+8B)*cos(x))
@test integrate((1+-1*sin(x))^-4*(A+B*sin(x)), x) == :((105+-105*sin(x))^-1*(-8B+6A)*cos(x)+1//7*(1+-1*sin(x))^-4*(A+B)*cos(x)+1//35*(1+-1*sin(x))^-3*(-4B+3A)*cos(x)+1//105*(1+-1*sin(x))^-2*(-8B+6A)*cos(x))
@test integrate((c+d*sin(e+f*x))^5//2*(a+a*sin(e+f*x)), x) == :(-2//7*a*f^-1*(c+d*sin(e+f*x))^5//2*cos(e+f*x)+-2//35*a*f^-1*(c+d*sin(e+f*x))^3//2*(5c+7d)*cos(e+f*x)+-2//105*a*f^-1*(c+d*sin(e+f*x))^(1/2)*(15*c^2+25*d^2+56*c*d)*cos(e+f*x)+2//105*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(15*c^3+63*d^3+145*c*d^2+161*d*c^2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-2//105*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(15*c^2+25*d^2+56*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^3//2*(a+a*sin(e+f*x)), x) == :(-2//5*a*f^-1*(c+d*sin(e+f*x))^3//2*cos(e+f*x)+-2//15*a*f^-1*(c+d*sin(e+f*x))^(1/2)*(3c+5d)*cos(e+f*x)+2//15*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(3*c^2+9*d^2+20*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-2//15*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(3c+5d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^(1/2)*(a+a*sin(e+f*x)), x) == :(-2//3*a*f^-1*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-2//3*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2//3*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+3d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^-1//2*(a+a*sin(e+f*x)), x) == :(2*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-2*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+-1d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^-3//2*(a+a*sin(e+f*x)), x) == :(-2*a*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)+2*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-2*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^-5//2*(a+a*sin(e+f*x)), x) == :(-2*a*f^-1*(c+d*sin(e+f*x))^-3//2*(3c+3d)^-1*cos(e+f*x)+-2*a*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*(-3d+3c)^-1*(c+-3d)*cos(e+f*x)+2//3*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-2*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^(1/2)*(-3d+3c)^-1*(c+-3d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^-7//2*(a+a*sin(e+f*x)), x) == :(-2*a*f^-1*(c+d*sin(e+f*x))^-5//2*(5c+5d)^-1*cos(e+f*x)+-2*a*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-3//2*(-15d+15c)^-1*(-5d+3c)*cos(e+f*x)+-2//15*a*f^-1*(c+d)^-3*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(3*c^2+9*d^2+-20*c*d)*cos(e+f*x)+2*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*(-15d+15c)^-1*(-5d+3c)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-2//15*a*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-3*(c+-1d)^-2*(c+d*sin(e+f*x))^(1/2)*(3*c^2+9*d^2+-20*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^5//2, x) == :(-2//9*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^7//2*cos(e+f*x)+4//63*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^5//2*(c+-9d)*cos(e+f*x)+4//315*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(-75*d^3+5*c^3+-141*c*d^2+-45*d*c^2)*cos(e+f*x)+4//315*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^3//2*(-56*d^2+5*c*(c+-9d))*cos(e+f*x)+-4//315*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(-168*d^4+5*c^4+-435*c*d^3+-381*c^2*d^2+-45*d*c^3)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//315*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(-75*d^3+5*c^3+-141*c*d^2+-45*d*c^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^3//2, x) == :(-2//7*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^5//2*cos(e+f*x)+4//35*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(c^2+-10*d^2+-7*c*d)*cos(e+f*x)+4//35*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^3//2*(c+-7d)*cos(e+f*x)+-4//35*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+3d)*(c^2+-7*d^2+-10*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//35*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(c^2+-10*d^2+-7*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(1/2), x) == :(-2//5*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^3//2*cos(e+f*x)+4//15*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(c+-5d)*cos(e+f*x)+-4//15*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-12*d^2+-5*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//15*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+-5d)*(c^2+-1*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^-1//2, x) == :(-2//3*a^2*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-4//3*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+-3d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//3*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+-1d)*(c+-2d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^-3//2, x) == :(-4*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+-1d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2*a^2*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*(c+-1d)*cos(e+f*x)+4*c*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^-5//2, x) == :(-4//3*a^2*d^-1*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*(c+3d)*cos(e+f*x)+2//3*a^2*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-3//2*(c+-1d)*cos(e+f*x)+-4//3*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^(1/2)*(c+3d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//3*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*(c+2d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^-7//2, x) == :(-4//15*a^2*d^-1*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-3//2*(c+5d)*cos(e+f*x)+2//5*a^2*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-5//2*(c+-1d)*cos(e+f*x)+-4*a^2*d^-1*f^-1*(c+d)^-3*(c+d*sin(e+f*x))^-1//2*(-15d+15c)^-1*(c^2+-12*d^2+5*c*d)*cos(e+f*x)+4//15*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*(c+5d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-4*a^2*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-3*(c+d*sin(e+f*x))^(1/2)*(-15d+15c)^-1*(c^2+-12*d^2+5*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^5//2, x) == :(-2//11*d^-1*f^-1*(c+d*sin(e+f*x))^7//2*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+-4//693*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(4*c^4+315*d^4+-33*d*c^3+177*c^2*d^2+561*c*d^3)*cos(e+f*x)+-4//693*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^3//2*(4*c^3+231*d^3+-33*d*c^2+182*c*d^2)*cos(e+f*x)+-4//693*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^5//2*(4*c^2+189*d^2+-33*c*d)*cos(e+f*x)+8//99*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^7//2*(c+-6d)*cos(e+f*x)+-4//693*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(4*c^4+315*d^4+-33*d*c^3+177*c^2*d^2+561*c*d^3)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//693*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+3d)*(4*c^4+231*d^4+-45*d*c^3+309*c^2*d^2+525*c*d^3)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^3//2, x) == :(-2//9*d^-1*f^-1*(c+d*sin(e+f*x))^5//2*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+-4//315*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(4*c^3+165*d^3+-27*d*c^2+114*c*d^2)*cos(e+f*x)+-4//315*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^3//2*(4*c^2+119*d^2+-27*c*d)*cos(e+f*x)+8//63*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^5//2*(c+-5d)*cos(e+f*x)+4//315*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(4*c^4+357*d^4+-27*d*c^3+111*c^2*d^2+579*c*d^3)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-4//315*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(4*c^3+165*d^3+-27*d*c^2+114*c*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^(1/2), x) == :(-2//7*d^-1*f^-1*(c+d*sin(e+f*x))^3//2*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+-4//105*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(4*c^2+65*d^2+-21*c*d)*cos(e+f*x)+8//35*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^3//2*(c+-4d)*cos(e+f*x)+4//105*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(4*c^3+147*d^3+-21*d*c^2+62*c*d^2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-4//105*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(4*c^2+65*d^2+-21*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-1//2, x) == :(-2//5*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(a^3+a^3*sin(e+f*x))*cos(e+f*x)+8//15*a^3*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(c+-3d)*cos(e+f*x)+4//15*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(4*c^2+27*d^2+-15*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-4//15*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+-1d)*(4*c^2+15*d^2+-11*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-3//2, x) == :(d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*(a^3+a^3*sin(e+f*x))*(-2d+2c)*cos(e+f*x)+-4//3*a^3*d^-2*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^(1/2)*(-1d+2c)*cos(e+f*x)+-4//3*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^(1/2)*(-3*d^2+4*c^2+-5*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//3*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+-1d)*(-5d+4c)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-5//2, x) == :(1//3*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-3//2*(a^3+a^3*sin(e+f*x))*(-2d+2c)*cos(e+f*x)+4//3*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^(1/2)*(-3*d^2+4*c^2+5*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+8//3*a^3*d^-2*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*(c+-1d)*(c+2d)*cos(e+f*x)+-4//3*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*(c+-1d)*(4c+5d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-7//2, x) == :(-4//15*a^3*d^-2*f^-1*(c+d)^-3*(c+d*sin(e+f*x))^-1//2*(4*c^2+27*d^2+15*c*d)*cos(e+f*x)+1//5*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-5//2*(a^3+a^3*sin(e+f*x))*(-2d+2c)*cos(e+f*x)+-4//15*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-3*(c+d*sin(e+f*x))^(1/2)*(4*c^2+27*d^2+15*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//15*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*(4*c^2+15*d^2+11*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+8//15*a^3*d^-2*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-3//2*(c+-1d)*(c+3d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^-9//2, x) == :(-4//105*a^3*d^-2*f^-1*(c+d)^-3*(c+d*sin(e+f*x))^-3//2*(4*c^2+65*d^2+21*c*d)*cos(e+f*x)+1//7*d^-1*f^-1*(c+d)^-1*(c+d*sin(e+f*x))^-7//2*(a^3+a^3*sin(e+f*x))*(-2d+2c)*cos(e+f*x)+-4*a^3*d^-2*f^-1*(c+d)^-4*(c+d*sin(e+f*x))^-1//2*(-105d+105c)^-1*(-147*d^3+4*c^3+21*d*c^2+62*c*d^2)*cos(e+f*x)+4//105*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-3*(c+d*sin(e+f*x))^-1//2*(4*c^2+65*d^2+21*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+8//35*a^3*d^-2*f^-1*(c+d)^-2*(c+d*sin(e+f*x))^-5//2*(c+-1d)*(c+4d)*cos(e+f*x)+-4*a^3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-4*(c+d*sin(e+f*x))^(1/2)*(-105d+105c)^-1*(-147*d^3+4*c^3+21*d*c^2+62*c*d^2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^5//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^3//2*(c+-1d)*cos(e+f*x)+-1//3*a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(3*c^2+9*d^2+-20*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//3*d*a^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(-5d+3c)*cos(e+f*x)+1//3*a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(-5d+3c)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^3//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^(1/2)*(c+-1d)*cos(e+f*x)+a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1*a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+-3d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^(1/2), x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-1*a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-1//2, x) == :(a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1*f^-1*(a+a*sin(e+f*x))^-1*(c+-1d)^-1*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-1*a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+-1d)^-1*(c+d*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-3//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*(c+-1d)^-1*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)+a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+-1d)^-1*(c+d*sin(e+f*x))^-1//2*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1*d*a^-1*f^-1*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(c+3d)*cos(e+f*x)+-1*a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^(1/2)*(c+3d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-5//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-1*(c+-1d)^-1*(c+d*sin(e+f*x))^-3//2*cos(e+f*x)+-1//3*d*a^-1*f^-1*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-3//2*(3c+5d)*cos(e+f*x)+-1//3*d*a^-1*f^-1*(c+d)^-2*(c+-1d)^-3*(c+d*sin(e+f*x))^-1//2*(3*c^2+9*d^2+20*c*d)*cos(e+f*x)+-1//3*a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-2*(c+-1d)^-3*(c+d*sin(e+f*x))^(1/2)*(3*c^2+9*d^2+20*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//3*a^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(3c+5d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^5//2, x) == :(-1//3*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^3//2*(c+-1d)*cos(e+f*x)+-1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-12*d^2+5*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+d*sin(e+f*x))^(1/2)*(c+-1d)*(c+5d)*cos(e+f*x)+1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+5d)*(c^2+-1*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^3//2, x) == :(-1//3*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^(1/2)*(c+-1d)*cos(e+f*x)+-1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+3d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+d*sin(e+f*x))^(1/2)*(c+3d)*cos(e+f*x)+1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+d)*(c+2d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^(1/2), x) == :(-1//3*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*c*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+-1d)^-1*(c+d*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*c*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^-1*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-1//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^(1/2)*(-3d+3c)^-1*cos(e+f*x)+-1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+-1d)^-2*(c+d*sin(e+f*x))^(1/2)*(c+-3d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^(1/2)*(c+-3d)*cos(e+f*x)+1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+-1d)^-1*(c+d*sin(e+f*x))^-1//2*(c+-2d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-3//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-1//2*(-3d+3c)^-1*cos(e+f*x)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(c+-5d)*cos(e+f*x)+1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(c+-5d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*d*a^-2*f^-1*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-1//2*(c^2+-12*d^2+-5*c*d)*cos(e+f*x)+-1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^(1/2)*(c^2+-12*d^2+-5*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-5//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-3//2*(-3d+3c)^-1*cos(e+f*x)+-1//3*a^-2*f^-1*(1+sin(e+f*x))^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-3//2*(c+-7d)*cos(e+f*x)+-1//3*d*a^-2*f^-1*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-3//2*(c^2+-10*d^2+-7*c*d)*cos(e+f*x)+1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-1//2*(c^2+-10*d^2+-7*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*d*a^-2*f^-1*(c+d)^-2*(c+-1d)^-4*(c+d*sin(e+f*x))^-1//2*(c+3d)*(c^2+-7*d^2+-10*c*d)*cos(e+f*x)+-1//3*a^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-2*(c+-1d)^-4*(c+d*sin(e+f*x))^(1/2)*(c+3d)*(c^2+-7*d^2+-10*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^5//2, x) == :(-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^3//2*(c+-1d)*cos(e+f*x)+-1//30*f^-1*(c+d*sin(e+f*x))^(1/2)*(a^3+a^3*sin(e+f*x))^-1*(4*c^2+27*d^2+15*c*d)*cos(e+f*x)+-1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(4*c^2+27*d^2+15*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^(1/2)*(c+3d)*(-2c+2d)*cos(e+f*x)+1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+d)*(4*c^2+15*d^2+11*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^3//2, x) == :(-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^(1/2)*(c+-1d)*cos(e+f*x)+-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(a^3+a^3*sin(e+f*x))^-1*(-30d+30c)^-1*(-3*d^2+4*c^2+5*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+d*sin(e+f*x))^(1/2)*(2c+4d)*cos(e+f*x)+-1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+-1d)^-1*(c+d*sin(e+f*x))^(1/2)*(-3*d^2+4*c^2+5*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c+d)*(4c+5d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^(1/2), x) == :(-1//5*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-1//30*f^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^(1/2)*(a^3+a^3*sin(e+f*x))^-1*(-3*d^2+4*c^2+-5*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)^-1*(c+d*sin(e+f*x))^(1/2)*(-1d+2c)*cos(e+f*x)+-1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+-1d)^-2*(c+d*sin(e+f*x))^(1/2)*(-3*d^2+4*c^2+-5*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+-1d)^-1*(c+d*sin(e+f*x))^-1//2*(c+d)*(-5d+4c)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-1//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^(1/2)*(-5d+5c)^-1*cos(e+f*x)+-1//30*f^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^(1/2)*(a^3+a^3*sin(e+f*x))^-1*(4*c^2+27*d^2+-15*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)^-2*(c+d*sin(e+f*x))^(1/2)*(-6d+2c)*cos(e+f*x)+-1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+-1d)^-3*(c+d*sin(e+f*x))^(1/2)*(4*c^2+27*d^2+-15*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(4*c^2+15*d^2+-11*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-3//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-1//2*(-5d+5c)^-1*cos(e+f*x)+-1//30*f^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-1//2*(a^3+a^3*sin(e+f*x))^-1*(4*c^2+65*d^2+-21*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(-8d+2c)*cos(e+f*x)+1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+-1d)^-3*(c+d*sin(e+f*x))^-1//2*(4*c^2+65*d^2+-21*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//30*d*a^-3*f^-1*(c+d)^-1*(c+-1d)^-4*(c+d*sin(e+f*x))^-1//2*(4*c^3+147*d^3+-21*d*c^2+62*c*d^2)*cos(e+f*x)+-1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-1*(c+-1d)^-4*(c+d*sin(e+f*x))^(1/2)*(4*c^3+147*d^3+-21*d*c^2+62*c*d^2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-5//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-3//2*(-5d+5c)^-1*cos(e+f*x)+-1//30*f^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-3//2*(a^3+a^3*sin(e+f*x))^-1*(4*c^2+119*d^2+-27*c*d)*cos(e+f*x)+-1//15*a^-1*f^-1*(a+a*sin(e+f*x))^-2*(c+-1d)^-2*(c+d*sin(e+f*x))^-3//2*(-10d+2c)*cos(e+f*x)+-1//30*d*a^-3*f^-1*(c+d)^-1*(c+-1d)^-4*(c+d*sin(e+f*x))^-3//2*(4*c^3+165*d^3+-27*d*c^2+114*c*d^2)*cos(e+f*x)+-1//30*d*a^-3*f^-1*(c+d)^-2*(c+-1d)^-5*(c+d*sin(e+f*x))^-1//2*(4*c^4+357*d^4+-27*d*c^3+111*c^2*d^2+579*c*d^3)*cos(e+f*x)+-1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d)^-2*(c+-1d)^-5*(c+d*sin(e+f*x))^(1/2)*(4*c^4+357*d^4+-27*d*c^3+111*c^2*d^2+579*c*d^3)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//30*a^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d)^-1*(c+-1d)^-4*(c+d*sin(e+f*x))^-1//2*(4*c^3+165*d^3+-27*d*c^2+114*c*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^3, x) == :(-2//7*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3*cos(e+f*x)+-12//35*a^-1*d^2*f^-1*(a+a*sin(e+f*x))^3//2*(c+d)*cos(e+f*x)+-4//35*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)*(7*d^2+15*c^2+10*c*d)*cos(e+f*x)+-1//35*d*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)*(-8d+40c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^2, x) == :(-2//5*a^-1*d^2*f^-1*(a+a*sin(e+f*x))^3//2*cos(e+f*x)+-2//15*a*f^-1*(a+a*sin(e+f*x))^-1//2*(7*d^2+15*c^2+10*c*d)*cos(e+f*x)+-1//15*d*f^-1*(a+a*sin(e+f*x))^(1/2)*(-4d+20c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x)), x) == :(-2//3*d*f^-1*(a+a*sin(e+f*x))^(1/2)*cos(e+f*x)+-2//3*a*f^-1*(a+a*sin(e+f*x))^-1//2*(d+3c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2), x) == :(-2*a*f^-1*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-1, x) == :(-2*a^(1/2)*d^-1//2*f^-1*(c+d)^-1//2*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-2, x) == :(-1*a^(1/2)*d^-1//2*f^-1*(c+d)^-3//2*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-3, x) == :(-3//4*a^(1/2)*d^-1//2*f^-1*(c+d)^-5//2*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-2*(2c+2d)^-1*cos(e+f*x)+-3//4*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^3, x) == :(-2//9*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^4*cos(e+f*x)+1//105*d*f^-1*(a+a*sin(e+f*x))^3//2*(c+d)*(-68d+4c)*cos(e+f*x)+2//63*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3*(c+-17d)*cos(e+f*x)+8//315*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)*(c+-17d)*(-1d+5c)*cos(e+f*x)+4//315*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)*(c+-17d)*(7*d^2+15*c^2+10*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^2, x) == :(-8//105*a^2*f^-1*(a+a*sin(e+f*x))^-1//2*(19*d^2+35*c^2+42*c*d)*cos(e+f*x)+-2//7*a^-1*d^2*f^-1*(a+a*sin(e+f*x))^5//2*cos(e+f*x)+-2//105*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(19*d^2+35*c^2+42*c*d)*cos(e+f*x)+-1//35*d*f^-1*(a+a*sin(e+f*x))^3//2*(-4d+28c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x)), x) == :(-2//5*d*f^-1*(a+a*sin(e+f*x))^3//2*cos(e+f*x)+-8//15*a^2*f^-1*(a+a*sin(e+f*x))^-1//2*(3d+5c)*cos(e+f*x)+-2//15*a*f^-1*(a+a*sin(e+f*x))^(1/2)*(3d+5c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2, x) == :(-8//3*a^2*f^-1*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)+-2//3*a*f^-1*(a+a*sin(e+f*x))^(1/2)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-1, x) == :(-2*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)+2*a^3//2*d^-3//2*f^-1*(c+d)^-1//2*(c+-1d)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-2, x) == :(-1*a^3//2*d^-3//2*f^-1*(c+d)^-3//2*(c+3d)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^-1*(c+-1d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-3, x) == :(-1//4*a^3//2*d^-3//2*f^-1*(c+d)^-5//2*(c+7d)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+(1/2)*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^-2*(c+-1d)*cos(e+f*x)+-1//4*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-1*(c+7d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^3, x) == :(-4//1155*a*f^-1*(a+a*sin(e+f*x))^3//2*(c+d)*(3*c^2+355*d^2+-38*c*d)*cos(e+f*x)+-2//11*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^4*cos(e+f*x)+-2//693*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3*(3*c^2+355*d^2+-38*c*d)*cos(e+f*x)+2//99*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^4*(-23d+3c)*cos(e+f*x)+-8//3465*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)*(-1d+5c)*(3*c^2+355*d^2+-38*c*d)*cos(e+f*x)+-4//3465*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)*(3*c^2+355*d^2+-38*c*d)*(7*d^2+15*c^2+10*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^2, x) == :(-64//315*a^3*f^-1*(a+a*sin(e+f*x))^-1//2*(13*d^2+21*c^2+30*c*d)*cos(e+f*x)+-16//315*a^2*f^-1*(a+a*sin(e+f*x))^(1/2)*(13*d^2+21*c^2+30*c*d)*cos(e+f*x)+-2//9*a^-1*d^2*f^-1*(a+a*sin(e+f*x))^7//2*cos(e+f*x)+-2//105*a*f^-1*(a+a*sin(e+f*x))^3//2*(13*d^2+21*c^2+30*c*d)*cos(e+f*x)+-1//63*d*f^-1*(a+a*sin(e+f*x))^5//2*(-4d+36c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x)), x) == :(-2//7*d*f^-1*(a+a*sin(e+f*x))^5//2*cos(e+f*x)+-64//105*a^3*f^-1*(a+a*sin(e+f*x))^-1//2*(5d+7c)*cos(e+f*x)+-16//105*a^2*f^-1*(a+a*sin(e+f*x))^(1/2)*(5d+7c)*cos(e+f*x)+-2//35*a*f^-1*(a+a*sin(e+f*x))^3//2*(5d+7c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2, x) == :(-64//15*a^3*f^-1*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)+-16//15*a^2*f^-1*(a+a*sin(e+f*x))^(1/2)*cos(e+f*x)+-2//5*a*f^-1*(a+a*sin(e+f*x))^3//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-1, x) == :(-2//3*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*cos(e+f*x)+-2*a^5//2*d^-5//2*f^-1*(c+d)^-1//2*(c+-1d)^2*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+2//3*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(-7d+3c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-2, x) == :(a^5//2*d^-5//2*f^-1*(c+d)^-3//2*(c+-1d)*(3c+5d)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(d+3c)*cos(e+f*x)+a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-1*(c+-1d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-3, x) == :(-1//4*a^5//2*d^-5//2*f^-1*(c+d)^-5//2*(3*c^2+19*d^2+10*c*d)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+(1/2)*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-2*(c+-1d)*cos(e+f*x)+3//4*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-1*(c+-1d)*(c+3d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^3*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+-4//15*d*f^-1*(a+a*sin(e+f*x))^-1//2*(7*d^2+21*c^2+-12*c*d)*cos(e+f*x)+-2//5*d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^2*cos(e+f*x)+-1//15*a^-1*d^2*f^-1*(a+a*sin(e+f*x))^(1/2)*(-2d+18c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^2, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+-2//3*a^-1*d^2*f^-1*(a+a*sin(e+f*x))^(1/2)*cos(e+f*x)+-1//3*d*f^-1*(a+a*sin(e+f*x))^-1//2*(-4d+12c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x)), x) == :(-2*d*f^-1*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)+-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2, x) == :(-1*2^(1/2)*a^-1//2*f^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+2*a^-1//2*d^(1/2)*f^-1*(c+d)^-1//2*(c+-1d)^-1*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-2, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^-2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-1*cos(e+f*x)+a^-1//2*d^(1/2)*f^-1*(c+d)^-3//2*(c+-1d)^-2*(d+3c)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-3, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^-3*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-2*(-2*d^2+2*c^2)^-1*cos(e+f*x)+1//4*d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-2*(d+7c)*cos(e+f*x)+1//4*a^-1//2*d^(1/2)*f^-1*(c+d)^-5//2*(c+-1d)^-3*(7*d^2+15*c^2+10*c*d)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^3, x) == :(-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^2*(c+-1d)*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^2*(c+11d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+1//3*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(3*c^2+13*d^2+-24*c*d)*cos(e+f*x)+1//6*a^-2*d^2*f^-1*(a+a*sin(e+f*x))^(1/2)*(-7d+3c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^2, x) == :(-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)*(c+d*sin(e+f*x))*cos(e+f*x)+(1/2)*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-5d)*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)*(c+7d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x)), x) == :(-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+3d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2, x) == :(-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-1, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3//2*(-2d+2c)^-1*cos(e+f*x)+-2*a^-3//2*d^3//2*f^-1*(c+d)^-1//2*(c+-1d)^-2*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^-2*(c+-5d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-1*(-2d+2c)^-1*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^-3*(c+-9d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+-1*a^-3//2*d^3//2*f^-1*(c+d)^-3//2*(c+-1d)^-3*(3d+5c)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1//2*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-1*(c+3d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-3, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-2*(-2d+2c)^-1*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^-4*(c+-13d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+-1//4*a^-3//2*d^3//2*f^-1*(c+d)^-5//2*(c+-1d)^-4*(19*d^2+35*c^2+42*c*d)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1//2*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-2*(c+2d)*cos(e+f*x)+-1//4*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+-1d)^-3*(c+d*sin(e+f*x))^-1*(c+7d)*(d+2c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^3, x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^2*(c+-1d)*cos(e+f*x)+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)^2*(3c+13d)*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(-3d+3c)*(c^2+25*d^2+6*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+1//4*a^-2*d^2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+-9d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^2, x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1d)*(c+d*sin(e+f*x))*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(3*c^2+19*d^2+10*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+3d)*(-3d+3c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x)), x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+-1d)*cos(e+f*x)+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(3c+5d)*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(3c+5d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-5//2, x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*cos(e+f*x)+-3//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*cos(e+f*x)+-3//32*2^(1/2)*a^-5//2*f^-1*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-1, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-5//2*(-4d+4c)^-1*cos(e+f*x)+2*a^-5//2*d^5//2*f^-1*(c+d)^-1//2*(c+-1d)^-3*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)^-2*(-11d+3c)*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(c+-1d)^-3*(3*c^2+43*d^2+-14*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-1*(-4d+4c)^-1*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(c+-1d)^-4*(3*c^2+115*d^2+-22*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+a^-5//2*d^5//2*f^-1*(c+d)^-3//2*(c+-1d)^-4*(5d+7c)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)^-2*(c+d*sin(e+f*x))^-1*(-15d+3c)*cos(e+f*x)+-1//16*d*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-1*(c+-7d)*(3c+5d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-3, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-2*(-4d+4c)^-1*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(c+-1d)^-5*(3*c^2+219*d^2+-30*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)^-2*(c+d*sin(e+f*x))^-2*(-19d+3c)*cos(e+f*x)+3//4*a^-5//2*d^5//2*f^-1*(c+d)^-5//2*(c+-1d)^-5*(13*d^2+21*c^2+30*c*d)*arctanh(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d)^-1//2*cos(e+f*x))+-1//16*d*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-2*(-31*d^2+3*c^2+-20*c*d)*cos(e+f*x)+-3//16*d*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+-1d)^-4*(c+d*sin(e+f*x))^-1*(c+3d)*(c^2+-7*d^2+-10*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^5//2, x) == :(-5//8*a^(1/2)*d^-1//2*f^-1*(c+d)^3*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//3*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^5//2*cos(e+f*x)+-5//8*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^2*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-5//12*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*(c+d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^3//2, x) == :(-3//4*a^(1/2)*d^-1//2*f^-1*(c+d)^2*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//2*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*cos(e+f*x)+-3//4*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(1/2), x) == :(-1*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-1*a^(1/2)*d^-1//2*f^-1*(c+d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-1//2, x) == :(-2*a^(1/2)*d^-1//2*f^-1*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-3//2, x) == :(-2*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-5//2, x) == :(-2*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-3//2*(3c+3d)^-1*cos(e+f*x)+-4//3*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-7//2, x) == :(-2*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-5//2*(5c+5d)^-1*cos(e+f*x)+-16//15*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-3*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)+-8//15*a*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-3//2*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^5//2, x) == :(-1//4*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^7//2*cos(e+f*x)+5//64*a^3//2*d^-3//2*f^-1*(c+d)^3*(c+-15d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+1//24*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^5//2*(c+-15d)*cos(e+f*x)+5//64*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^2*(c+d*sin(e+f*x))^(1/2)*(c+-15d)*cos(e+f*x)+5//96*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*(c+d)*(c+-15d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^3//2, x) == :(-1//3*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^5//2*cos(e+f*x)+1//8*a^3//2*d^-3//2*f^-1*(c+d)^2*(c+-11d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+1//12*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*(c+-11d)*cos(e+f*x)+1//8*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+d)*(c+-11d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^(1/2), x) == :(-1//2*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*cos(e+f*x)+1//4*a^3//2*d^-3//2*f^-1*(c+d)*(c+-7d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+1//4*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+-7d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-1//2, x) == :(a^3//2*d^-3//2*f^-1*(c+-3d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-3//2, x) == :(-2*a^3//2*d^-3//2*f^-1*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+2*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*(c+-1d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-5//2, x) == :(-2//3*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*(c+5d)*cos(e+f*x)+2//3*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^-3//2*(c+-1d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-7//2, x) == :(-4//15*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-3*(c+d*sin(e+f*x))^-1//2*(c+9d)*cos(e+f*x)+-2//15*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-3//2*(c+9d)*cos(e+f*x)+2//5*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^-5//2*(c+-1d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-9//2, x) == :(-16//105*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-4*(c+d*sin(e+f*x))^-1//2*(c+13d)*cos(e+f*x)+-8//105*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-3*(c+d*sin(e+f*x))^-3//2*(c+13d)*cos(e+f*x)+-2//35*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-5//2*(c+13d)*cos(e+f*x)+2//7*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^-7//2*(c+-1d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^5//2, x) == :(-1//5*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^7//2*cos(e+f*x)+-1//128*a^5//2*d^-5//2*f^-1*(c+d)^3*(3*c^2+283*d^2+-34*c*d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//240*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^5//2*(3*c^2+283*d^2+-34*c*d)*cos(e+f*x)+3//40*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^7//2*(c+-7d)*cos(e+f*x)+-1//128*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^2*(c+d*sin(e+f*x))^(1/2)*(3*c^2+283*d^2+-34*c*d)*cos(e+f*x)+-1//192*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*(c+d)*(3*c^2+283*d^2+-34*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^3//2, x) == :(-1//4*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^5//2*cos(e+f*x)+-1//64*a^5//2*d^-5//2*f^-1*(c+d)^2*(3*c^2+163*d^2+-26*c*d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//96*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*(3*c^2+163*d^2+-26*c*d)*cos(e+f*x)+1//24*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^5//2*(-17d+3c)*cos(e+f*x)+-1//64*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+d)*(3*c^2+163*d^2+-26*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^(1/2), x) == :(-1//3*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^3//2*cos(e+f*x)+-1//8*a^5//2*d^-5//2*f^-1*(c+d)*(c^2+25*d^2+-6*c*d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//8*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+25*d^2+-6*c*d)*cos(e+f*x)+1//12*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*(-13d+3c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-1//2, x) == :(-1//4*a^5//2*d^-5//2*f^-1*(3*c^2+19*d^2+-10*c*d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//2*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+3//4*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+-3d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-3//2, x) == :(a^5//2*d^-5//2*f^-1*(-5d+3c)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+d*sin(e+f*x))^(1/2)*(-1d+3c)*cos(e+f*x)+2*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-1//2*(c+-1d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-5//2, x) == :(-2*a^5//2*d^-5//2*f^-1*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+2//3*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-3//2*(c+-1d)*cos(e+f*x)+2//3*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-1//2*(c+-1d)*(3c+7d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-7//2, x) == :(-2//15*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-3*(c+d*sin(e+f*x))^-1//2*(3*c^2+43*d^2+14*c*d)*cos(e+f*x)+2//5*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-5//2*(c+-1d)*cos(e+f*x)+2//15*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-3//2*(c+-1d)*(3c+11d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-9//2, x) == :(-4//105*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-4*(c+d*sin(e+f*x))^-1//2*(3*c^2+115*d^2+22*c*d)*cos(e+f*x)+-2//105*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-3*(c+d*sin(e+f*x))^-3//2*(3*c^2+115*d^2+22*c*d)*cos(e+f*x)+2//7*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-7//2*(c+-1d)*cos(e+f*x)+6//35*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-5//2*(c+-1d)*(c+5d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^5//2*(c+d*sin(e+f*x))^-11//2, x) == :(-16//315*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-5*(c+d*sin(e+f*x))^-1//2*(c^2+73*d^2+10*c*d)*cos(e+f*x)+-8//315*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-4*(c+d*sin(e+f*x))^-3//2*(c^2+73*d^2+10*c*d)*cos(e+f*x)+-2//105*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-3*(c+d*sin(e+f*x))^-5//2*(c^2+73*d^2+10*c*d)*cos(e+f*x)+2//9*a^2*d^-1*f^-1*(a+a*sin(e+f*x))^(1/2)*(c+d)^-1*(c+d*sin(e+f*x))^-9//2*(c+-1d)*cos(e+f*x)+2//63*a^3*d^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+d*sin(e+f*x))^-7//2*(c+-1d)*(3c+19d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^5//2, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^5//2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//2*d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2*cos(e+f*x)+-1//4*a^-1//2*d^(1/2)*f^-1*(7*d^2+15*c^2+-10*c*d)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//4*d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*(-1d+7c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^3//2, x) == :(-1*d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^3//2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1*a^-1//2*d^(1/2)*f^-1*(-1d+3c)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2), x) == :(-2*a^-1//2*d^(1/2)*f^-1*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^(1/2)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^-1//2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-3//2, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^-3//2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+2*d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-1*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-5//2, x) == :(-1*2^(1/2)*a^-1//2*f^-1*(c+-1d)^-5//2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+2*d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-3//2*(-3*d^2+3*c^2)^-1*cos(e+f*x)+2//3*d*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-2*(d+5c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^5//2, x) == :(-1*a^-3//2*d^3//2*f^-1*(-3d+5c)*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^3//2*(c+-1d)*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^3//2*(c+9d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+(1/2)*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c+-3d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^3//2, x) == :(-2*a^-3//2*d^3//2*f^-1*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2)*(c+-1d)*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^(1/2)*(c+5d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2), x) == :(-1//2*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^-1//2*(c+d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-1//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2)*(-2d+2c)^-1*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^-3//2*(c+-3d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-3//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-1//2*(-2d+2c)^-1*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^-5//2*(c+-7d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//2*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(c+5d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-5//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-3//2*(-2d+2c)^-1*cos(e+f*x)+-1//4*2^(1/2)*a^-3//2*f^-1*(c+-1d)^-7//2*(c+-11d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//6*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+-1d)^-2*(c+d*sin(e+f*x))^-3//2*(3c+7d)*cos(e+f*x)+-1//6*d*a^-1*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+-1d)^-3*(c+d*sin(e+f*x))^-1//2*(3*c^2+19*d^2+38*c*d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^5//2, x) == :(-2*a^-5//2*d^5//2*f^-1*arctan(a^(1/2)*d^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^3//2*(c+-1d)*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(c+-1d)^(1/2)*(3*c^2+43*d^2+14*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2)*(c+-1d)*(3c+11d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^3//2, x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^(1/2)*(c+-1d)*cos(e+f*x)+-3//32*2^(1/2)*a^-5//2*f^-1*(c+d)^2*(c+-1d)^-1//2*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2)*(3c+7d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^(1/2), x) == :(-1//4*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)^-1*(c+d*sin(e+f*x))^(1/2)*(-1d+3c)*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(c+-1d)^-3//2*(c+d)*(-5d+3c)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x)))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-1//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^(1/2)*(-4d+4c)^-1*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(c+-1d)^-5//2*(3*c^2+19*d^2+-10*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)^-2*(c+d*sin(e+f*x))^(1/2)*(-9d+3c)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-3//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-1//2*(-4d+4c)^-1*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(c+-1d)^-7//2*(3*c^2+75*d^2+-18*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)^-2*(c+d*sin(e+f*x))^-1//2*(-13d+3c)*cos(e+f*x)+-1//16*d*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-1//2*(c+-7d)*(3c+7d)*cos(e+f*x))
@test integrate((a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-5//2, x) == :(-1*f^-1*(a+a*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-3//2*(-4d+4c)^-1*cos(e+f*x)+-1//32*2^(1/2)*a^-5//2*f^-1*(c+-1d)^-9//2*(3*c^2+163*d^2+-26*c*d)*arctanh((1/2)*2^(1/2)*a^(1/2)*(a+a*sin(e+f*x))^-1//2*(c+-1d)^(1/2)*(c+d*sin(e+f*x))^-1//2*cos(e+f*x))+-1//16*a^-1*f^-1*(a+a*sin(e+f*x))^-3//2*(c+-1d)^-2*(c+d*sin(e+f*x))^-3//2*(-17d+3c)*cos(e+f*x)+-1//48*d*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-1*(c+-1d)^-3*(c+d*sin(e+f*x))^-3//2*(-95*d^2+9*c^2+-54*c*d)*cos(e+f*x)+-1//48*d*a^-2*f^-1*(a+a*sin(e+f*x))^-1//2*(c+d)^-2*(c+-1d)^-4*(c+d*sin(e+f*x))^-1//2*(-299*d^3+9*c^3+-493*c*d^2+-57*d*c^2)*cos(e+f*x))
@test integrate((1+sin(e+f*x))^m*(3+3*sin(e+f*x))^(-1+-1m), x) == :(-1*3^(-1+-1m)*f^-1*(1+sin(e+f*x))^-1*cos(e+f*x))
@test integrate((1+sin(e+f*x))^m*(3+-3*sin(e+f*x))^(-1+-1m), x) == :(f^-1*(1+2m)^-1*(1+sin(e+f*x))^m*(3+-3*sin(e+f*x))^(-1+-1m)*cos(e+f*x))
@test integrate((3+3*sin(e+f*x))^(-1+-1m)*(a+a*sin(e+f*x))^m, x) == :(-1*f^-1*(3+3*sin(e+f*x))^(-1+-1m)*(a+a*sin(e+f*x))^m*cos(e+f*x))
@test integrate((3+-3*sin(e+f*x))^(-1+-1m)*(a+a*sin(e+f*x))^m, x) == :(f^-1*(1+2m)^-1*(3+-3*sin(e+f*x))^(-1+-1m)*(a+a*sin(e+f*x))^m*cos(e+f*x))
@test integrate((-3+3*sin(e+f*x))^(-1+-1m)*(a+a*sin(e+f*x))^m, x) == :(f^-1*(1+2m)^-1*(-3+3*sin(e+f*x))^(-1+-1m)*(a+a*sin(e+f*x))^m*cos(e+f*x))
@test integrate((-3+-3*sin(e+f*x))^(-1+-1m)*(a+a*sin(e+f*x))^m, x) == :(-1*f^-1*(-3+-3*sin(e+f*x))^(-1+-1m)*(a+a*sin(e+f*x))^m*cos(e+f*x))
@test integrate((c+d*sin(e+f*x))^3*(a+b*sin(e+f*x)), x) == :(x*(a*c^3+3//8*b*d^3+3//2*a*c*d^2+3//2*b*d*c^2)+-1//6*f^-1*(3*b*(c^3+4*c*d^2)+4*a*d*(d^2+4*c^2))*cos(e+f*x)+-1//4*b*f^-1*(c+d*sin(e+f*x))^3*cos(e+f*x)+-1//12*f^-1*(c+d*sin(e+f*x))^2*(3*b*c+4*a*d)*cos(e+f*x)+-1//24*d*f^-1*(6*b*c^2+9*b*d^2+20*a*c*d)*cos(e+f*x)*sin(e+f*x))
@test integrate((c+d*sin(e+f*x))^2*(a+b*sin(e+f*x)), x) == :(x*((1/2)*a*(d^2+2*c^2)+b*c*d)+-1//3*f^-1*(2*b*(c^2+d^2)+6*a*c*d)*cos(e+f*x)+-1//3*b*f^-1*(c+d*sin(e+f*x))^2*cos(e+f*x)+-1//6*d*f^-1*(2*b*c+3*a*d)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))*(c+d*sin(e+f*x)), x) == :(x*(a*c+(1/2)*b*d)+-1*f^-1*(a*d+b*c)*cos(e+f*x)+-1//2*b*d*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate(a+b*sin(e+f*x), x) == :(a*x+-1*b*f^-1*cos(e+f*x))
@test integrate((c+d*sin(e+f*x))^-1*(a+b*sin(e+f*x)), x) == :(b*x*d^-1+-1*d^-1*f^-1*(c^2+-1*d^2)^-1//2*(-2*a*d+2*b*c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((c+d*sin(e+f*x))^-2*(a+b*sin(e+f*x)), x) == :(f^-1*(c^2+-1*d^2)^-3//2*(-2*b*d+2*a*c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-1*(b*c+-1*a*d)*cos(e+f*x))
@test integrate((c+d*sin(e+f*x))^-3*(a+b*sin(e+f*x)), x) == :(-1*f^-1*(c^2+-1*d^2)^-5//2*(-1*a*(d^2+2*c^2)+3*b*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+(1/2)*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-2*(-1*b*(c^2+2*d^2)+3*a*c*d)*cos(e+f*x)+-1*f^-1*(c+d*sin(e+f*x))^-2*(-2*d^2+2*c^2)^-1*(b*c+-1*a*d)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^3, x) == :(x*((1/2)*a^2*(2*c^3+3*c*d^2)+1//8*c*b^2*(4*c^2+9*d^2)+3//4*a*b*d*(d^2+4*c^2))+-1//30*d^-1*f^-1*(-1*b^2*(-16*d^4+3*c^4+-52*c^2*d^2)+20*a^2*d^2*(d^2+4*c^2)+30*a*b*c*d*(c^2+4*d^2))*cos(e+f*x)+-1//120*f^-1*(-1*b^2*(6*c^3+-71*c*d^2)+100*c*a^2*d^2+30*a*b*d*(2*c^2+3*d^2))*cos(e+f*x)*sin(e+f*x)+-1//5*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^4*cos(e+f*x)+-1//60*d^-1*f^-1*(c+d*sin(e+f*x))^2*(d^2*(16*b^2+20*a^2)+-3*b*c*(b*c+-10*a*d))*cos(e+f*x)+1//20*b*d^-1*f^-1*(c+d*sin(e+f*x))^3*(b*c+-10*a*d)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^2, x) == :(x*((1/2)*a^2*(d^2+2*c^2)+1//8*b^2*(3*d^2+4*c^2)+2*a*b*c*d)+-1//6*b^-1*f^-1*(-1*a^3*d^2+4*a*b^2*(2*d^2+3*c^2)+8*c*d*b^3+8*b*c*d*a^2)*cos(e+f*x)+-1//24*f^-1*(3*b^2*(3*d^2+4*c^2)+2*a*d*(-1*a*d+8*b*c))*cos(e+f*x)*sin(e+f*x)+-1//4*b^-1*d^2*f^-1*(a+b*sin(e+f*x))^3*cos(e+f*x)+-1//12*d*b^-1*f^-1*(a+b*sin(e+f*x))^2*(-1*a*d+8*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x)), x) == :(x*(c*a^2+(1/2)*c*b^2+a*b*d)+-1//3*f^-1*(2*d*a^2+2*d*b^2+6*a*b*c)*cos(e+f*x)+-1//3*d*f^-1*(a+b*sin(e+f*x))^2*cos(e+f*x)+-1//6*b*f^-1*(2*a*d+3*b*c)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^2, x) == :(x*(a^2+(1/2)*b^2)+-2*a*b*f^-1*cos(e+f*x)+-1//2*b^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^-1, x) == :(-1*b*x*d^-2*(b*c+-2*a*d)+-1*b^2*d^-1*f^-1*cos(e+f*x)+2*d^-2*f^-1*(c^2+-1*d^2)^-1//2*(b*c+-1*a*d)^2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^-2, x) == :(x*b^2*d^-2+d^-1*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+-1*d^-2*f^-1*(c^2+-1*d^2)^-3//2*(b*(c^2+-2*d^2)+a*c*d)*(-2*a*d+2*b*c)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^-3, x) == :(-1*f^-1*(c^2+-1*d^2)^-5//2*(-1*a^2*(d^2+2*c^2)+-1*b^2*(c^2+2*d^2)+6*a*b*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+(1/2)*d^-1*f^-1*(c+d*sin(e+f*x))^-2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+-1//2*d^-1*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-2*(b*c+-1*a*d)*(b*(c^2+-4*d^2)+3*a*c*d)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^-4, x) == :(-1*f^-1*(c^2+-1*d^2)^-7//2*(-1*a^2*(2*c^3+3*c*d^2)+-1*c*b^2*(c^2+4*d^2)+2*a*b*d*(d^2+4*c^2))*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+1//3*d^-1*f^-1*(c+d*sin(e+f*x))^-3*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+1//6*d^-1*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-3*(-1*b^2*(c^4+-6*d^4+-10*c^2*d^2)+a^2*d^2*(4*d^2+11*c^2)+-1*a*b*(4*d*c^3+26*c*d^3))*cos(e+f*x)+-1//6*d^-1*f^-1*(c+d*sin(e+f*x))^-2*(c^2+-1*d^2)^-2*(b*c+-1*a*d)*(b*(c^2+-6*d^2)+5*a*c*d)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^2, x) == :(x*((1/2)*a^3*(d^2+2*c^2)+3//4*c*d*b^3+3//8*a*b^2*(3*d^2+4*c^2)+3*b*c*d*a^2)+-1//30*b^-1*f^-1*(-3*a^4*d^2+4*b^4*(4*d^2+5*c^2)+4*a^2*b^2*(13*d^2+20*c^2)+30*b*c*d*a^3+120*a*c*d*b^3)*cos(e+f*x)+-1//120*f^-1*(-6*a^3*d^2+a*b^2*(71*d^2+100*c^2)+90*c*d*b^3+60*b*c*d*a^2)*cos(e+f*x)*sin(e+f*x)+-1//5*b^-1*d^2*f^-1*(a+b*sin(e+f*x))^4*cos(e+f*x)+-1//60*b^-1*f^-1*(a+b*sin(e+f*x))^2*(4*b^2*(4*d^2+5*c^2)+3*a*d*(-1*a*d+10*b*c))*cos(e+f*x)+-1//20*d*b^-1*f^-1*(a+b*sin(e+f*x))^3*(-1*a*d+10*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x)), x) == :(x*(c*a^3+3//8*d*b^3+3//2*a*c*b^2+3//2*b*d*a^2)+-1//6*f^-1*(3*d*a^3+4*c*b^3+12*a*d*b^2+16*b*c*a^2)*cos(e+f*x)+-1//4*d*f^-1*(a+b*sin(e+f*x))^3*cos(e+f*x)+-1//12*f^-1*(a+b*sin(e+f*x))^2*(3*a*d+4*b*c)*cos(e+f*x)+-1//24*b*f^-1*(6*d*a^2+9*d*b^2+20*a*b*c)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^3, x) == :((1/2)*a*x*(2*a^2+3*b^2)+-2//3*b*f^-1*(b^2+4*a^2)*cos(e+f*x)+-1//3*b*f^-1*(a+b*sin(e+f*x))^2*cos(e+f*x)+-5//6*a*b^2*f^-1*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-1, x) == :(-1//2*b*x*d^-3*(-1*b^2*(d^2+2*c^2)+-6*a^2*d^2+6*a*b*c*d)+(1/2)*b^2*d^-2*f^-1*(-5*a*d+2*b*c)*cos(e+f*x)+-2*d^-3*f^-1*(c^2+-1*d^2)^-1//2*(b*c+-1*a*d)^3*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//2*b^2*d^-1*f^-1*(a+b*sin(e+f*x))*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-2, x) == :(-1*x*b^2*d^-3*(-3*a*d+2*b*c)+b*d^-2*f^-1*(c^2+-1*d^2)^-1*(-1*a^2*d^2+-1*b^2*(-1*d^2+2*c^2)+2*a*b*c*d)*cos(e+f*x)+2*d^-3*f^-1*(c^2+-1*d^2)^-3//2*(b*c+-1*a*d)^2*(-3*b*d^2+2*b*c^2+a*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+d^-1*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*(a+b*sin(e+f*x))*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-3, x) == :(x*b^3*d^-3+-1*d^-3*f^-1*(c^2+-1*d^2)^-5//2*(b^3*(2*c^5+-5*c^3*d^2+6*c*d^4)+-1*a^3*d^3*(d^2+2*c^2)+-3*a*b^2*d^3*(c^2+2*d^2)+9*b*c*a^2*d^4)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+(1/2)*d^-1*f^-1*(c+d*sin(e+f*x))^-2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*(a+b*sin(e+f*x))*cos(e+f*x)+(1/2)*d^-2*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-2*(b*c+-1*a*d)^2*(-5*b*d^2+2*b*c^2+3*a*c*d)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-4, x) == :(-1*f^-1*(c^2+-1*d^2)^-7//2*(a*c+-1*b*d)*(-1*a^2*(2*c^2+3*d^2)+-1*b^2*(2*d^2+3*c^2)+10*a*b*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1//6*d^-2*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-3*(b*c+-1*a*d)*(b^2*(2*c^4+18*d^4+-5*c^2*d^2)+a^2*d^2*(4*d^2+11*c^2)+5*a*b*c*d*(c^2+-7*d^2))*cos(e+f*x)+1//3*d^-1*f^-1*(c+d*sin(e+f*x))^-3*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*(a+b*sin(e+f*x))*cos(e+f*x)+1//6*d^-2*f^-1*(c+d*sin(e+f*x))^-2*(c^2+-1*d^2)^-2*(b*c+-1*a*d)^2*(-7*b*d^2+2*b*c^2+5*a*c*d)*cos(e+f*x))
@test integrate((a+b*sin(x))^-1*(B*sin(x)+B*b*a^-1), x) == :(B*x*b^-1+-2*B*a^-1*b^-1*(a^2+-1*b^2)^(1/2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*x))))
@test integrate((a+b*sin(x))^-1*(B*sin(x)+B*a*b^-1), x) == :(B*x*b^-1)
@test integrate((b+a*sin(x))^-2*(a+b*sin(x)), x) == :(-1*(b+a*sin(x))^-1*cos(x))
@test integrate((2+sin(x))^-1*(2+-1*sin(x)), x) == :(-1x+4//3*x*3^(1/2)+8//3*3^(1/2)*arctan((2+3^(1/2)+sin(x))^-1*cos(x)))
@test integrate((a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^4, x) == :((1/2)*d*x*b^-4*(-2*a^3*d^3+4*c*b^3*(d^2+2*c^2)+-1*a*d*b^2*(d^2+12*c^2)+8*b*c*a^2*d^2)+2*b^-4*f^-1*(a^2+-1*b^2)^-1//2*(b*c+-1*a*d)^4*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+-1//3*b^-1*d^2*f^-1*(c+d*sin(e+f*x))^2*cos(e+f*x)+1//3*b^-3*d^2*f^-1*(-1*b^2*(2*d^2+17*c^2)+-3*a^2*d^2+12*a*b*c*d)*cos(e+f*x)+-1//6*b^-2*d^3*f^-1*(-3*a*d+8*b*c)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^3, x) == :(-1//2*d*x*b^-3*(-1*b^2*(d^2+6*c^2)+-2*a^2*d^2+6*a*b*c*d)+2*b^-3*f^-1*(a^2+-1*b^2)^-1//2*(b*c+-1*a*d)^3*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+-1//2*b^-1*d^2*f^-1*(c+d*sin(e+f*x))*cos(e+f*x)+-1//2*b^-2*d^2*f^-1*(-2*a*d+5*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^2, x) == :(d*x*b^-2*(-1*a*d+2*b*c)+-1*b^-1*d^2*f^-1*cos(e+f*x)+2*b^-2*f^-1*(a^2+-1*b^2)^-1//2*(b*c+-1*a*d)^2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x)), x) == :(d*x*b^-1+b^-1*f^-1*(a^2+-1*b^2)^-1//2*(-2*a*d+2*b*c)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^-1, x) == :(2*f^-1*(a^2+-1*b^2)^-1//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-1, x) == :(-2*d*f^-1*(c^2+-1*d^2)^-1//2*(b*c+-1*a*d)^-1*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+2*b*f^-1*(a^2+-1*b^2)^-1//2*(b*c+-1*a*d)^-1*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-2, x) == :(2*b^2*f^-1*(a^2+-1*b^2)^-1//2*(b*c+-1*a*d)^-2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+-1*d^2*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+2*d*f^-1*(c^2+-1*d^2)^-3//2*(b*c+-1*a*d)^-2*(-1*b*(-1*d^2+2*c^2)+a*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-3, x) == :(2*b^3*f^-1*(a^2+-1*b^2)^-1//2*(b*c+-1*a*d)^-3*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+d*f^-1*(c^2+-1*d^2)^-5//2*(b*c+-1*a*d)^-3*(-1*b^2*(2*d^4+6*c^4+-5*c^2*d^2)+-1*a^2*d^2*(d^2+2*c^2)+6*a*b*d*c^3)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+-1*d^2*f^-1*(c+d*sin(e+f*x))^-2*(c^2+-1*d^2)^-1*(-2*a*d+2*b*c)^-1*cos(e+f*x)+-1//2*d^2*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-2*(b*c+-1*a*d)^-2*(-2*b*d^2+5*b*c^2+-3*a*c*d)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^4, x) == :(-1//2*x*b^-4*d^2*(-1*b^2*(d^2+12*c^2)+-6*a^2*d^2+16*a*b*c*d)+2*b^-4*f^-1*(a^2+-1*b^2)^-3//2*(b*c+-1*a*d)^3*(-4*d*b^2+3*d*a^2+a*b*c)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+d*b^-3*f^-1*(a^2+-1*b^2)^-1*(-1*a*d+2*b*c)*(-1*b^2*(c^2+-2*d^2)+-3*a^2*d^2+2*a*b*c*d)*cos(e+f*x)+b^-1*f^-1*(a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+(1/2)*b^-2*d^2*f^-1*(a^2+-1*b^2)^-1*(-1*b^2*(-1*d^2+2*c^2)+-3*a^2*d^2+4*a*b*c*d)*cos(e+f*x)*sin(e+f*x))
@test integrate((a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^3, x) == :(x*b^-3*d^2*(-2*a*d+3*b*c)+d*b^-2*f^-1*(a^2+-1*b^2)^-1*(-1*b^2*(c^2+-1*d^2)+-2*a^2*d^2+2*a*b*c*d)*cos(e+f*x)+2*b^-3*f^-1*(a^2+-1*b^2)^-3//2*(b*c+-1*a*d)^2*(-3*d*b^2+2*d*a^2+a*b*c)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+b^-1*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^2*(c+d*sin(e+f*x))*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^2, x) == :(x*b^-2*d^2+b^-1*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+b^-2*f^-1*(a^2+-1*b^2)^-3//2*(-2*a*d+2*b*c)*(d*a^2+-2*d*b^2+a*b*c)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x)), x) == :(f^-1*(a^2+-1*b^2)^-3//2*(-2*b*d+2*a*c)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-1*(b*c+-1*a*d)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-2, x) == :(2*a*f^-1*(a^2+-1*b^2)^-3//2*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+b*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-1*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-1, x) == :(2*d^2*f^-1*(c^2+-1*d^2)^-1//2*(b*c+-1*a*d)^-2*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+b^2*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+2*b*f^-1*(a^2+-1*b^2)^-3//2*(b*c+-1*a*d)^-2*(d*b^2+-2*d*a^2+a*b*c)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x))))
@test integrate((a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-2, x) == :(2*b^2*f^-1*(a^2+-1*b^2)^-3//2*(b*c+-1*a*d)^-3*(-3*d*a^2+2*d*b^2+a*b*c)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+2*d^2*f^-1*(c^2+-1*d^2)^-3//2*(b*c+-1*a*d)^-3*(-2*b*d^2+3*b*c^2+-1*a*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+b^2*f^-1*(a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-1*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+d*f^-1*(c+d*sin(e+f*x))^-1*(a^2+-1*b^2)^-1*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^-2*(a^2*d^2+b^2*(c^2+-2*d^2))*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-3, x) == :(-1*d^2*f^-1*(c^2+-1*d^2)^-5//2*(b*c+-1*a*d)^-4*(-3*b^2*(2*d^4+4*c^4+-5*c^2*d^2)+-1*a^2*d^2*(d^2+2*c^2)+2*a*b*c*d*(-1*d^2+4*c^2))*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+2*b^3*f^-1*(a^2+-1*b^2)^-3//2*(b*c+-1*a*d)^-4*(-4*d*a^2+3*d*b^2+a*b*c)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+b^2*f^-1*(a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+-1*f^-1*(c+d*sin(e+f*x))^-1*(c^2+-1*d^2)^-2*(-2*b^2+2*a^2)^-1*(b*c+-1*a*d)^-3*(-1*b^3*(6*d^5+-11*c^2*d^3+2*d*c^4)+3*c*a^3*d^4+-1*b*a^2*d^3*(-4*d^2+7*c^2)+-3*a*c*b^2*d^4)*cos(e+f*x)+d*f^-1*(c+d*sin(e+f*x))^-2*(c^2+-1*d^2)^-1*(-2*b^2+2*a^2)^-1*(b*c+-1*a*d)^-2*(a^2*d^2+b^2*(-3*d^2+2*c^2))*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3*(c+d*sin(e+f*x))^5, x) == :(-1//2*x*b^-5*d^3*(-1*b^2*(d^2+20*c^2)+-12*a^2*d^2+30*a*b*c*d)+b^-5*f^-1*(a^2+-1*b^2)^-5//2*(b*c+-1*a*d)^3*(b^4*(c^2+20*d^2)+12*a^4*d^2+a^2*b^2*(-29*d^2+2*c^2)+-12*a*c*d*b^3+6*b*c*d*a^3)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+-1//2*d*b^-4*f^-1*(a^2+-1*b^2)^-2*(-12*a^5*d^4+a*b^4*(-6*d^4+6*c^4+43*c^2*d^2)+-1*c*d*b^5*(-10*d^2+17*c^2)+-1*a^3*b^2*d^2*(-21*d^2+16*c^2)+30*b*c*a^4*d^3+-1*c*d*a^2*b^3*(4*c^2+55*d^2))*cos(e+f*x)+(1/2)*b^-1*f^-1*(a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^3*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+(1/2)*b^-3*d^2*f^-1*(a^2+-1*b^2)^-2*(-6*a^4*d^3+d*b^4*(-1*d^2+8*c^2)+d*a^2*b^2*(c^2+10*d^2)+-1*a*c*b^3*(3*c^2+16*d^2)+7*b*c*a^3*d^2)*cos(e+f*x)*sin(e+f*x)+(1/2)*b^-2*f^-1*(a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^2*(a^2+-1*b^2)^-2*(b*c+-1*a*d)^2*(-7*d*b^2+4*d*a^2+3*a*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3*(c+d*sin(e+f*x))^4, x) == :(x*b^-4*d^3*(-3*a*d+4*b*c)+b^-4*f^-1*(a^2+-1*b^2)^-5//2*(b*c+-1*a*d)^2*(b^4*(c^2+12*d^2)+6*a^4*d^2+a^2*b^2*(-15*d^2+2*c^2)+-10*a*c*d*b^3+4*b*c*d*a^3)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+(1/2)*b^-3*d^2*f^-1*(a^2+-1*b^2)^-1*(-1*b^2*(c^2+-2*d^2)+-3*a^2*d^2+2*a*b*c*d)*cos(e+f*x)+(1/2)*b^-1*f^-1*(a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+3//2*b^-3*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*(b*c+-1*a*d)^3*(d*a^2+-2*d*b^2+a*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3*(c+d*sin(e+f*x))^3, x) == :(x*b^-3*d^3+b^-3*f^-1*(a^2+-1*b^2)^-5//2*(b*c+-1*a*d)*(b^4*(c^2+6*d^2)+2*a^4*d^2+a^2*b^2*(-5*d^2+2*c^2)+-8*a*c*d*b^3+2*b*c*d*a^3)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+(1/2)*b^-1*f^-1*(a+b*sin(e+f*x))^-2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^2*(c+d*sin(e+f*x))*cos(e+f*x)+(1/2)*b^-2*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*(b*c+-1*a*d)^2*(-5*d*b^2+2*d*a^2+3*a*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3*(c+d*sin(e+f*x))^2, x) == :(-1*f^-1*(a^2+-1*b^2)^-5//2*(-1*a^2*(d^2+2*c^2)+-1*b^2*(c^2+2*d^2)+6*a*b*c*d)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+(1/2)*b^-1*f^-1*(a+b*sin(e+f*x))^-2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+(1/2)*b^-1*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*(b*c+-1*a*d)*(d*a^2+-4*d*b^2+3*a*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3*(c+d*sin(e+f*x)), x) == :(f^-1*(a^2+-1*b^2)^-5//2*(c*b^2+2*c*a^2+-3*a*b*d)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+f^-1*(a+b*sin(e+f*x))^-2*(-2*b^2+2*a^2)^-1*(b*c+-1*a*d)*cos(e+f*x)+(1/2)*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*(-1*d*a^2+-2*d*b^2+3*a*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3, x) == :(f^-1*(a^2+-1*b^2)^-5//2*(b^2+2*a^2)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+b*f^-1*(a+b*sin(e+f*x))^-2*(-2*b^2+2*a^2)^-1*cos(e+f*x)+3//2*a*b*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-1, x) == :(-2*d^3*f^-1*(c^2+-1*d^2)^-1//2*(b*c+-1*a*d)^-3*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+b^2*f^-1*(a+b*sin(e+f*x))^-2*(-2*b^2+2*a^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+-1*b*f^-1*(a^2+-1*b^2)^-5//2*(b*c+-1*a*d)^-3*(-1*b^4*(c^2+2*d^2)+-6*a^4*d^2+-1*a^2*b^2*(-5*d^2+2*c^2)+6*b*c*d*a^3)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+(1/2)*b^2*f^-1*(a+b*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*(b*c+-1*a*d)^-2*(-5*d*a^2+2*d*b^2+3*a*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-2, x) == :(-1*b^2*f^-1*(a^2+-1*b^2)^-5//2*(b*c+-1*a*d)^-4*(-1*b^4*(c^2+6*d^2)+-12*a^4*d^2+-1*a^2*b^2*(-15*d^2+2*c^2)+-2*a*c*d*b^3+8*b*c*d*a^3)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+-2*d^3*f^-1*(c^2+-1*d^2)^-3//2*(b*c+-1*a*d)^-4*(-3*b*d^2+4*b*c^2+-1*a*c*d)*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+b^2*f^-1*(a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-1*(-2*b^2+2*a^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+-1//2*d*f^-1*(c+d*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^-3*(2*a^4*d^3+-2*d*b^4*(-3*d^2+2*c^2)+d*a^2*b^2*(-11*d^2+7*c^2)+-3*a*c*b^3*(c^2+-1*d^2))*cos(e+f*x)+3//2*b^2*f^-1*(a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*(b*c+-1*a*d)^-2*(d*b^2+-2*d*a^2+a*b*c)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3*(c+d*sin(e+f*x))^-3, x) == :(-1*b^3*f^-1*(a^2+-1*b^2)^-5//2*(b*c+-1*a*d)^-5*(-1*b^4*(c^2+12*d^2)+-20*a^4*d^2+-1*a^2*b^2*(-29*d^2+2*c^2)+-4*a*c*d*b^3+10*b*c*d*a^3)*arctan((a^2+-1*b^2)^-1//2*(b+a*tan((1/2)*e+(1/2)*f*x)))+-1*d^3*f^-1*(c^2+-1*d^2)^-5//2*(b*c+-1*a*d)^-5*(b^2*(12*d^4+20*c^4+-29*c^2*d^2)+a^2*d^2*(d^2+2*c^2)+-1*a*b*(-4*c*d^3+10*d*c^3))*arctan((c^2+-1*d^2)^-1//2*(d+c*tan((1/2)*e+(1/2)*f*x)))+b^2*f^-1*(a+b*sin(e+f*x))^-2*(c+d*sin(e+f*x))^-2*(-2*b^2+2*a^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+(1/2)*b^2*f^-1*(a+b*sin(e+f*x))^-1*(c+d*sin(e+f*x))^-2*(a^2+-1*b^2)^-2*(b*c+-1*a*d)^-2*(-7*d*a^2+4*d*b^2+3*a*b*c)*cos(e+f*x)+-1//2*d*f^-1*(c+d*sin(e+f*x))^-2*(a^2+-1*b^2)^-2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^-3*(a^4*d^3+-1*d*b^4*(-6*d^2+5*c^2)+-3*a*c*b^3*(c^2+-1*d^2)+2*d*a^2*b^2*(-5*d^2+4*c^2))*cos(e+f*x)+3//2*d*f^-1*(c+d*sin(e+f*x))^-1*(a^2+-1*b^2)^-2*(c^2+-1*d^2)^-2*(b*c+-1*a*d)^-4*(c*a^5*d^4+d*b^5*(2*c^4+4*d^4+-7*c^2*d^2)+-1*b*a^4*(-2*d^5+3*c^2*d^3)+a*c*b^4*(c^4+2*d^4+-2*c^2*d^2)+-1*d*a^2*b^3*(3*c^4+7*d^4+-12*c^2*d^2)+-2*c*a^3*b^2*d^4)*cos(e+f*x))
@test integrate((c+d*sin(e+f*x))^5//2*(a+b*sin(e+f*x)), x) == :(-2//7*b*f^-1*(c+d*sin(e+f*x))^5//2*cos(e+f*x)+-1//35*f^-1*(c+d*sin(e+f*x))^3//2*(10*b*c+14*a*d)*cos(e+f*x)+1//105*f^-1*(c+d*sin(e+f*x))^(1/2)*(-50*b*d^2+-30*b*c^2+-112*a*c*d)*cos(e+f*x)+1//105*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(30*b*c^3+126*a*d^3+290*b*c*d^2+322*a*d*c^2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//105*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(-2*d^2+2*c^2)*(15*b*c^2+25*b*d^2+56*a*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^3//2*(a+b*sin(e+f*x)), x) == :(-2//5*b*f^-1*(c+d*sin(e+f*x))^3//2*cos(e+f*x)+1//15*f^-1*(c+d*sin(e+f*x))^(1/2)*(-10*a*d+-6*b*c)*cos(e+f*x)+1//15*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(6*b*(c^2+3*d^2)+40*a*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//15*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(6*b*c+10*a*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^(1/2)*(a+b*sin(e+f*x)), x) == :(-2//3*b*f^-1*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+1//3*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(2*b*c+6*a*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-2//3*b*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^-1//2*(a+b*sin(e+f*x)), x) == :(-1*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(-2*a*d+2*b*c)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2*b*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^-3//2*(a+b*sin(e+f*x)), x) == :(f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-1*(-2*b*c+2*a*d)*cos(e+f*x)+2*b*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-1*(-2*a*d+2*b*c)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^-5//2*(a+b*sin(e+f*x)), x) == :(f^-1*(c+d*sin(e+f*x))^-3//2*(-3*d^2+3*c^2)^-1*(-2*b*c+2*a*d)*cos(e+f*x)+1//3*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-2*(-2*b*(c^2+3*d^2)+8*a*c*d)*cos(e+f*x)+1//3*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-1*(-2*a*d+2*b*c)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//3*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-2*(-2*b*(c^2+3*d^2)+8*a*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((c+d*sin(e+f*x))^-7//2*(a+b*sin(e+f*x)), x) == :(f^-1*(c+d*sin(e+f*x))^-5//2*(-5*d^2+5*c^2)^-1*(-2*b*c+2*a*d)*cos(e+f*x)+-1//15*f^-1*(c+d*sin(e+f*x))^-3//2*(c^2+-1*d^2)^-2*(6*b*c^2+10*b*d^2+-16*a*c*d)*cos(e+f*x)+-1//15*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-3*(-18*a*d^3+6*b*c^3+-46*a*d*c^2+58*b*c*d^2)*cos(e+f*x)+-1//15*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-3*(-18*a*d^3+6*b*c^3+-46*a*d*c^2+58*b*c*d^2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//15*d^-1*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-2*(6*b*c^2+10*b*d^2+-16*a*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^5//2, x) == :(-2//9*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^7//2*cos(e+f*x)+-1//315*d^-1*f^-1*(c+d*sin(e+f*x))^3//2*(2*d^2*(49*b^2+63*a^2)+-20*b*c*(b*c+-9*a*d))*cos(e+f*x)+1//315*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(4*b^2*(5*c^3+-57*c*d^2)+-336*c*a^2*d^2+-60*a*b*d*(3*c^2+5*d^2))*cos(e+f*x)+1//315*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(-2*b^2*(-147*d^4+10*c^4+-279*c^2*d^2)+42*a^2*d^2*(9*d^2+23*c^2)+60*a*b*d*(3*c^3+29*c*d^2))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//63*b*d^-1*f^-1*(c+d*sin(e+f*x))^5//2*(b*c+-9*a*d)*cos(e+f*x)+1//315*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(-4*d^2+4*c^2)*(5*b^2*c^3+-84*c*a^2*d^2+-75*a*b*d^3+-57*c*b^2*d^2+-45*a*b*d*c^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^3//2, x) == :(-2//7*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^5//2*cos(e+f*x)+1//105*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(-2*d^2*(25*b^2+35*a^2)+12*b*c*(b*c+-7*a*d))*cos(e+f*x)+1//105*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(-4*b^2*(3*c^3+-41*c*d^2)+280*c*a^2*d^2+84*a*b*d*(c^2+3*d^2))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//35*b*d^-1*f^-1*(c+d*sin(e+f*x))^3//2*(b*c+-7*a*d)*cos(e+f*x)+-1//105*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(-2*d^2+2*c^2)*(-1*b^2*(-25*d^2+6*c^2)+35*a^2*d^2+42*a*b*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(1/2), x) == :(-2//5*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^3//2*cos(e+f*x)+1//15*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(2*d^2*(9*b^2+15*a^2)+-4*b*c*(b*c+-5*a*d))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+4//15*b*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(b*c+-5*a*d)*cos(e+f*x)+4//15*b*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(b*c+-5*a*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^-1//2, x) == :(-2//3*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*cos(e+f*x)+1//3*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(2*d^2*(b^2+3*a^2)+4*b*c*(b*c+-3*a*d))*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-4//3*b*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(b*c+-3*a*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^-3//2, x) == :(2*d^-1*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-1*(2*d^2*(a^2+-1*b^2)+4*b^2*c^2+-4*a*b*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-4*b*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(b*c+-1*a*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^-5//2, x) == :(2//3*d^-1*f^-1*(c+d*sin(e+f*x))^-3//2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+-1//3*d^-1*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-2*(b*(c^2+-3*d^2)+2*a*c*d)*(-4*a*d+4*b*c)*cos(e+f*x)+1//3*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-1*(-2*a^2*d^2+2*b^2*(-3*d^2+2*c^2)+4*a*b*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-2*(b*(c^2+-3*d^2)+2*a*c*d)*(-4*a*d+4*b*c)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^-7//2, x) == :(1//15*d^-1*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-3*(-2*b^2*(-15*d^4+2*c^4+-19*c^2*d^2)+-2*a*b*(6*d*c^3+58*c*d^3)+2*a^2*d^2*(9*d^2+23*c^2))*cos(e+f*x)+2//5*d^-1*f^-1*(c+d*sin(e+f*x))^-5//2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*cos(e+f*x)+-1//15*d^-1*f^-1*(c+d*sin(e+f*x))^-3//2*(c^2+-1*d^2)^-2*(b*(c^2+-5*d^2)+4*a*c*d)*(-4*a*d+4*b*c)*cos(e+f*x)+1//15*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-3*(-2*b^2*(-15*d^4+2*c^4+-19*c^2*d^2)+-2*a*b*(6*d*c^3+58*c*d^3)+2*a^2*d^2*(9*d^2+23*c^2))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+1//15*d^-2*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-2*(b*(c^2+-5*d^2)+4*a*c*d)*(-4*a*d+4*b*c)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^5//2, x) == :(-1//3465*d^-2*f^-1*(c+d*sin(e+f*x))^3//2*(10*b^3*(8*c^3+67*c*d^2)+1386*a^3*d^3+-66*a*d*b^2*(-49*d^2+10*c^2)+2970*b*c*a^2*d^2)*cos(e+f*x)+1//3465*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(-10*b^3*(8*c^4+135*d^4+57*c^2*d^2)+-3696*c*a^3*d^3+-990*b*a^2*d^2*(3*c^2+5*d^2)+132*a*d*b^2*(5*c^3+-57*c*d^2))*cos(e+f*x)+-2//11*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^7//2*(a+b*sin(e+f*x))*cos(e+f*x)+1//3465*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(10*b^3*(8*c^5+51*c^3*d^2+741*c*d^4)+462*a^3*d^3*(9*d^2+23*c^2)+-66*a*d*b^2*(-147*d^4+10*c^4+-279*c^2*d^2)+990*b*c*a^2*d^2*(3*c^2+29*d^2))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2//693*b*d^-2*f^-1*(c+d*sin(e+f*x))^5//2*(-1*b^2*(8*c^2+81*d^2)+-297*a^2*d^2+66*a*b*c*d)*cos(e+f*x)+8//99*b^2*d^-2*f^-1*(c+d*sin(e+f*x))^7//2*(b*c+-6*a*d)*cos(e+f*x)+-1//3465*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(-2*d^2+2*c^2)*(5*b^3*(8*c^4+135*d^4+57*c^2*d^2)+1848*c*a^3*d^3+-66*a*d*b^2*(5*c^3+-57*c*d^2)+495*b*a^2*d^2*(3*c^2+5*d^2))*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^3//2, x) == :(1//315*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(-210*a^3*d^3+-2*b^3*(8*c^3+39*c*d^2)+-378*b*c*a^2*d^2+18*a*d*b^2*(-25*d^2+6*c^2))*cos(e+f*x)+-2//9*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^5//2*(a+b*sin(e+f*x))*cos(e+f*x)+1//315*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(2*b^3*(8*c^4+147*d^4+33*c^2*d^2)+-2*a*b^2*(-738*c*d^3+54*d*c^3)+840*c*a^3*d^3+378*b*a^2*d^2*(c^2+3*d^2))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2//315*b*d^-2*f^-1*(c+d*sin(e+f*x))^3//2*(-1*b^2*(8*c^2+49*d^2)+-189*a^2*d^2+54*a*b*c*d)*cos(e+f*x)+8//63*b^2*d^-2*f^-1*(c+d*sin(e+f*x))^5//2*(b*c+-5*a*d)*cos(e+f*x)+-1//315*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(-2*d^2+2*c^2)*(b^3*(8*c^3+39*c*d^2)+105*a^3*d^3+-9*a*d*b^2*(-25*d^2+6*c^2)+189*b*c*a^2*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^(1/2), x) == :(-2//7*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^3//2*(a+b*sin(e+f*x))*cos(e+f*x)+1//105*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(2*b^3*(8*c^3+19*c*d^2)+210*a^3*d^3+-42*a*d*b^2*(-9*d^2+2*c^2)+210*b*c*a^2*d^2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2//105*b*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(-1*b^2*(8*c^2+25*d^2)+-105*a^2*d^2+42*a*b*c*d)*cos(e+f*x)+8//35*b^2*d^-2*f^-1*(c+d*sin(e+f*x))^3//2*(b*c+-4*a*d)*cos(e+f*x)+2//105*b*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)*(-1*b^2*(8*c^2+25*d^2)+-105*a^2*d^2+42*a*b*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-1//2, x) == :(-2//5*b^2*d^-1*f^-1*(c+d*sin(e+f*x))^(1/2)*(a+b*sin(e+f*x))*cos(e+f*x)+-1//15*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(-30*a^3*d^3+2*b^3*(8*c^3+7*c*d^2)+-30*a*d*b^2*(d^2+2*c^2)+90*b*c*a^2*d^2)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+8//15*b^2*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(b*c+-3*a*d)*cos(e+f*x)+-2//15*b*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(-1*b^2*(8*c^2+9*d^2)+-45*a^2*d^2+30*a*b*c*d)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-3//2, x) == :(2*d^-1*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*(a+b*sin(e+f*x))*cos(e+f*x)+-2//3*b*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(-1*b^2*(d^2+8*c^2)+-9*a^2*d^2+18*a*b*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-1*(-6*a^3*d^3+2*b^3*(8*c^3+-5*c*d^2)+-18*a*d*b^2*(-1*d^2+2*c^2)+18*b*c*a^2*d^2)*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2//3*b*d^-2*f^-1*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-1*(-1*b^2*(-1*d^2+4*c^2)+-3*a^2*d^2+6*a*b*c*d)*cos(e+f*x))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-5//2, x) == :(1//3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-2*(2*b^3*(3*d^4+8*c^4+-15*c^2*d^2)+8*c*a^3*d^3+-6*b*a^2*d^2*(c^2+3*d^2)+-12*a*c*d*b^2*(c^2+-3*d^2))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2//3*d^-1*f^-1*(c+d*sin(e+f*x))^-3//2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*(a+b*sin(e+f*x))*cos(e+f*x)+8//3*d^-2*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-2*(b*c+-1*a*d)^2*(b*(c^2+-2*d^2)+a*c*d)*cos(e+f*x)+-1//3*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-1*(-2*a*d+2*b*c)*(b^2*(-9*d^2+8*c^2)+-1*a^2*d^2+2*a*b*c*d)*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-7//2, x) == :(-1//15*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-2*(-2*b^3*(8*c^4+15*d^4+-15*c^2*d^2)+16*c*a^3*d^3+-6*b*a^2*d^2*(3*c^2+5*d^2)+-12*a*c*d*b^2*(c^2+-5*d^2))*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+-1//15*d^-2*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-3*(-2*a*d+2*b*c)*(b^2*(8*c^4+45*d^4+-21*c^2*d^2)+a^2*d^2*(9*d^2+23*c^2)+2*a*b*d*(7*c^3+-39*c*d^2))*cos(e+f*x)+2//5*d^-1*f^-1*(c+d*sin(e+f*x))^-5//2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*(a+b*sin(e+f*x))*cos(e+f*x)+8//15*d^-2*f^-1*(c+d*sin(e+f*x))^-3//2*(c^2+-1*d^2)^-2*(b*c+-1*a*d)^2*(b*(c^2+-3*d^2)+2*a*c*d)*cos(e+f*x)+-1//15*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-3*(-2*a*d+2*b*c)*(b^2*(8*c^4+45*d^4+-21*c^2*d^2)+a^2*d^2*(9*d^2+23*c^2)+2*a*b*d*(7*c^3+-39*c*d^2))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^-9//2, x) == :(1//105*d^-2*f^-1*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-4*(-2*b^3*(8*c^6+105*d^6+-23*c^4*d^2+294*c^2*d^4)+-18*b*a^2*d^2*(5*c^4+21*d^4+102*c^2*d^2)+32*c*a^3*d^3*(11*c^2+13*d^2)+-12*a*c*d*b^2*(-133*d^4+3*c^4+-62*c^2*d^2))*cos(e+f*x)+-1//105*d^-2*f^-1*(c+d*sin(e+f*x))^-3//2*(c^2+-1*d^2)^-3*(-2*a*d+2*b*c)*(b^2*(8*c^4+105*d^4+-17*c^2*d^2)+a*b*(-218*c*d^3+26*d*c^3)+a^2*d^2*(25*d^2+71*c^2))*cos(e+f*x)+1//105*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^-1//2*(c+d*sin(e+f*x))^(1/2)*(c^2+-1*d^2)^-4*(-2*b^3*(8*c^6+105*d^6+-23*c^4*d^2+294*c^2*d^4)+-18*b*a^2*d^2*(5*c^4+21*d^4+102*c^2*d^2)+32*c*a^3*d^3*(11*c^2+13*d^2)+-12*a*c*d*b^2*(-133*d^4+3*c^4+-62*c^2*d^2))*Elliptic.E((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1)+2//7*d^-1*f^-1*(c+d*sin(e+f*x))^-7//2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^2*(a+b*sin(e+f*x))*cos(e+f*x)+8//35*d^-2*f^-1*(c+d*sin(e+f*x))^-5//2*(c^2+-1*d^2)^-2*(b*c+-1*a*d)^2*(b*(c^2+-4*d^2)+3*a*c*d)*cos(e+f*x)+1//105*d^-3*f^-1*((c+d)^-1*(c+d*sin(e+f*x)))^(1/2)*(c+d*sin(e+f*x))^-1//2*(c^2+-1*d^2)^-3*(-2*a*d+2*b*c)*(b^2*(8*c^4+105*d^4+-17*c^2*d^2)+a*b*(-218*c*d^3+26*d*c^3)+a^2*d^2*(25*d^2+71*c^2))*Elliptic.F((1/2)*e+-1//4*pi+(1/2)*f*x,2*d*(c+d)^-1))
@test integrate((a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-3//2, x) == :(f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-1//2*(c+-1d)^-1*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*(-2b+2a)*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+-1*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-1//2*(c+-1d)^-1*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*(-2b+2a)*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-5//2, x) == :(2*d*f^-1*(a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-3//2*(-3*d^2+3*c^2)^-1*cos(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-3//2*(c+-1d)^-2*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*(d+3c)*(-2b+2a)*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-3//2*(c+-1d)^-2*(b*c+-1*a*d)^-2*(c+d*sin(e+f*x))*(-2b+2a)*(-1*b*(d^2+3*c^2)+4*a*c*d)*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^3//2*(c+d*sin(e+f*x))^-5//2, x) == :(-1*f^-1*(a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-3//2*(-3*d^2+3*c^2)^-1*(-2*a*d+2*b*c)*cos(e+f*x)+-1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-3//2*(c+-1d)^-2*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*(-8b+8a)*(a*c+-1*b*d)*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-3//2*(c+-1d)^-2*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*(-2b+2a)*(a*(d+3c)+-1*b*(c+3d))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2, x) == :(2*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-1//2*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-3//2, x) == :(2*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-1//2*(c+-1d)^-1*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+d*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-1//2*(c+-1d)^-1*(b*c+-1*a*d)^-2*(c+d*sin(e+f*x))*(-2b+2a)*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-5//2, x) == :(-2*d^2*f^-1*(a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-3//2*(c^2+-1*d^2)^-1*(-3*a*d+3*b*c)^-1*cos(e+f*x)+-2//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-3//2*(c+-1d)^-2*(b*c+-1*a*d)^-2*(c+d*sin(e+f*x))*(-1*b*(-2*d^2+3*c^2+3*c*d)+a*d*(d+3c))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+-1//3*d*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(c+d)^-3//2*(c+-1d)^-2*(b*c+-1*a*d)^-3*(c+d*sin(e+f*x))*(-4b+4a)*(-1*b*(-1*d^2+3*c^2)+2*a*c*d)*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2), x) == :(f^-1*((a+b*sin(e+f*x))^-1*(c+-1d)^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+b*sin(e+f*x))^-1*(c+d)^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(a+-1b)^-1*(c+d)^(1/2)*(b*c+-1*a*d)^-1*(a+b*sin(e+f*x))*(-2d+2c)*Elliptic.E(arcsin((a+b)^(1/2)*(a+b*sin(e+f*x))^-1//2*(c+d)^-1//2*(c+d*sin(e+f*x))^(1/2)),(a+b)^-1*(c+-1d)^-1*(a+-1b)*(c+d))*sec(e+f*x)+2*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(a+-1b)^-1*(c+d)^-1//2*(b*c+-1*a*d)^-1*(c+-1d)*(c+d*sin(e+f*x))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-1//2, x) == :(2*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^(1/2)*(a+-1b)^-1*(c+d)^-1//2*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+2*b*f^-1*((a+b*sin(e+f*x))^-1*(c+-1d)^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+b*sin(e+f*x))^-1*(c+d)^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(a+-1b)^-1*(c+d)^(1/2)*(b*c+-1*a*d)^-2*(a+b*sin(e+f*x))*(c+-1d)*Elliptic.E(arcsin((a+b)^(1/2)*(a+b*sin(e+f*x))^-1//2*(c+d)^-1//2*(c+d*sin(e+f*x))^(1/2)),(a+b)^-1*(c+-1d)^-1*(a+-1b)*(c+d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-3//2, x) == :(2*b^2*f^-1*(a+b*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(c+d)^-1//2*(c+-1d)^-1*(b*c+-1*a*d)^-2*(c+d*sin(e+f*x))*(-2*a*d+2*b*(c+-2d))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+-1*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(c+d)^-1//2*(c+-1d)^-1*(b*c+-1*a*d)^-3*(c+d*sin(e+f*x))*(2*a^2*d^2+2*b^2*(c^2+-2*d^2))*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-5//2, x) == :(2*b^2*f^-1*(a+b*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-3//2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+2*d*f^-1*(a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-3//2*(c^2+-1*d^2)^-1*(-3*b^2+3*a^2)^-1*(b*c+-1*a*d)^-2*(a^2*d^2+b^2*(-4*d^2+3*c^2))*cos(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(c+d)^-3//2*(c+-1d)^-2*(b*c+-1*a*d)^-4*(c+d*sin(e+f*x))*(-2*b^3*(3*c^4+8*d^4+-15*c^2*d^2)+8*c*a^3*d^3+-8*a*c*b^2*d^3+-2*b*a^2*d^2*(-5*d^2+9*c^2))*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(c+d)^-3//2*(c+-1d)^-2*(b*c+-1*a*d)^-3*(c+d*sin(e+f*x))*(2*b^2*(3*c^3+8*d^3+-9*d*c^2+-6*c*d^2)+2*a^2*d^2*(d+3c)+-12*a*b*d*(c^2+-1*d^2))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^3//2, x) == :(f^-1*(a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2)*(-3*b^2+3*a^2)^-1*(-2*a*d+2*b*c)*cos(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(a+-1b)^-2*(c+d)^-1//2*(b*c+-1*a*d)^-1*(c+d*sin(e+f*x))*(-2d+2c)*(b*c+-1*a*d+-3*b*d+3*a*c)*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+1//3*f^-1*((a+b*sin(e+f*x))^-1*(c+-1d)^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+b*sin(e+f*x))^-1*(c+d)^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-3//2*(a+-1b)^-2*(c+d)^(1/2)*(b*c+-1*a*d)^-1*(a+b*sin(e+f*x))*(-8d+8c)*(a*c+-1*b*d)*Elliptic.E(arcsin((a+b)^(1/2)*(a+b*sin(e+f*x))^-1//2*(c+d)^-1//2*(c+d*sin(e+f*x))^(1/2)),(a+b)^-1*(c+-1d)^-1*(a+-1b)*(c+d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^(1/2), x) == :(2*b*f^-1*(a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2)*(-3*b^2+3*a^2)^-1*cos(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(a+-1b)^-2*(c+d)^-1//2*(b*c+-1*a*d)^-1*(c+-1d)*(c+d*sin(e+f*x))*(2b+6a)*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+1//3*f^-1*((a+b*sin(e+f*x))^-1*(c+-1d)^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+b*sin(e+f*x))^-1*(c+d)^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-3//2*(a+-1b)^-2*(c+d)^(1/2)*(b*c+-1*a*d)^-2*(a+b*sin(e+f*x))*(-2d+2c)*(-1*d*b^2+-3*d*a^2+4*a*b*c)*Elliptic.E(arcsin((a+b)^(1/2)*(a+b*sin(e+f*x))^-1//2*(c+d)^-1//2*(c+d*sin(e+f*x))^(1/2)),(a+b)^-1*(c+-1d)^-1*(a+-1b)*(c+d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-1//2, x) == :(2*b^2*f^-1*(a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^(1/2)*(-3*b^2+3*a^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(a+-1b)^-2*(c+d)^-1//2*(b*c+-1*a*d)^-2*(c+d*sin(e+f*x))*(-6*d*a^2+2*b^2*(c+2d)+6*a*b*(c+-1d))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+4//3*b*f^-1*((a+b*sin(e+f*x))^-1*(c+-1d)^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+b*sin(e+f*x))^-1*(c+d)^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-3//2*(a+-1b)^-2*(c+d)^(1/2)*(b*c+-1*a*d)^-3*(a+b*sin(e+f*x))*(c+-1d)*(d*b^2+-3*d*a^2+2*a*b*c)*Elliptic.E(arcsin((a+b)^(1/2)*(a+b*sin(e+f*x))^-1//2*(c+d)^-1//2*(c+d*sin(e+f*x))^(1/2)),(a+b)^-1*(c+-1d)^-1*(a+-1b)*(c+d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-3//2, x) == :(2*b^2*f^-1*(a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-1//2*(-3*b^2+3*a^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+8//3*b^2*f^-1*(a+b*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-1//2*(a^2+-1*b^2)^-2*(b*c+-1*a*d)^-2*(d*b^2+-2*d*a^2+a*b*c)*cos(e+f*x)+-1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(c+d)^-1//2*(c+-1d)^-1*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-3*(c+d*sin(e+f*x))*(-6*a^3*d^2+2*b^3*(c^2+8*d^2+-6*c*d)+-6*a*b^2*(c^2+-2*d^2)+6*b*d*a^2*(-3d+2c))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(c+d)^-1//2*(c+-1d)^-1*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-4*(c+d*sin(e+f*x))*(6*a^4*d^3+-2*d*b^4*(-8*d^2+5*c^2)+-8*a*c*b^3*(c^2+-1*d^2)+6*d*a^2*b^2*(-5*d^2+3*c^2))*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))
@test integrate((a+b*sin(e+f*x))^-5//2*(c+d*sin(e+f*x))^-5//2, x) == :(2*b^2*f^-1*(a+b*sin(e+f*x))^-3//2*(c+d*sin(e+f*x))^-3//2*(-3*b^2+3*a^2)^-1*(b*c+-1*a*d)^-1*cos(e+f*x)+4//3*b^2*f^-1*(a+b*sin(e+f*x))^-1//2*(c+d*sin(e+f*x))^-3//2*(a^2+-1*b^2)^-2*(b*c+-1*a*d)^-2*(-5*d*a^2+3*d*b^2+2*a*b*c)*cos(e+f*x)+-2//3*d*f^-1*(a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^-3//2*(a^2+-1*b^2)^-2*(c^2+-1*d^2)^-1*(b*c+-1*a*d)^-3*(a^4*d^3+-1*d*b^4*(-8*d^2+7*c^2)+d*a^2*b^2*(-13*d^2+11*c^2)+-4*a*c*b^3*(c^2+-1*d^2))*cos(e+f*x)+-1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(c+d)^-3//2*(c+-1d)^-2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-5*(c+d*sin(e+f*x))*(-8*b*a^4*(-2*d^5+3*c^2*d^3)+8*c*a^5*d^4+8*d*b^5*(2*c^4+4*d^4+-7*c^2*d^2)+-16*c*a^3*b^2*d^4+-8*d*a^2*b^3*(3*c^4+7*d^4+-12*c^2*d^2)+8*a*c*b^4*(c^4+2*d^4+-2*c^2*d^2))*Elliptic.E(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x)+-1//3*f^-1*((a+b)^-1*(c+d*sin(e+f*x))^-1*(1+-1*sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(-1*(a+-1b)^-1*(c+d*sin(e+f*x))^-1*(1+sin(e+f*x))*(b*c+-1*a*d))^(1/2)*(a+b)^-1//2*(c+d)^-3//2*(c+-1d)^-2*(a^2+-1*b^2)^-1*(b*c+-1*a*d)^-4*(c+d*sin(e+f*x))*(2*b^4*(c^4+-16*d^4+-9*d*c^3+12*c*d^3+16*c^2*d^2)+-6*a*b^3*(c^4+4*d^4+-5*c^2*d^2)+2*a^4*d^3*(d+3c)+-18*b*a^3*d^2*(c^2+-1*d^2)+2*d*a^2*b^2*(9*c^3+16*d^3+-18*d*c^2+-15*c*d^2))*Elliptic.F(arcsin((a+b)^-1//2*(a+b*sin(e+f*x))^(1/2)*(c+d)^(1/2)*(c+d*sin(e+f*x))^-1//2),(a+-1b)^-1*(c+d)^-1*(a+b)*(c+-1d))*sec(e+f*x))

max_stars_repo_path: test/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.jl