# Maple integration test file: "4 Trig functions\4.1 Sine\4.1.0 (a sin)^m (b trg)^n.txt"

lst:=[

# Integrands of the form (b Sin[c+d x])^m

# Integrands of the form (b Sin[c+d x])^n

# Integrands of the form Sin[c+d x]^n
[sin(a+b*x),x,1,-cos(a+b*x)/b],
[sin(a+b*x)^2,x,2,1/2*x-1/2*cos(a+b*x)*sin(a+b*x)/b],
[sin(a+b*x)^3,x,2,-cos(a+b*x)/b+1/3*cos(a+b*x)^3/b],
[sin(a+b*x)^4,x,3,3/8*x-3/8*cos(a+b*x)*sin(a+b*x)/b-1/4*cos(a+b*x)*sin(a+b*x)^3/b],
[sin(a+b*x)^5,x,2,-cos(a+b*x)/b+2/3*cos(a+b*x)^3/b-1/5*cos(a+b*x)^5/b],
[sin(a+b*x)^6,x,4,5/16*x-5/16*cos(a+b*x)*sin(a+b*x)/b-5/24*cos(a+b*x)*sin(a+b*x)^3/b-1/6*cos(a+b*x)*sin(a+b*x)^5/b],
[sin(a+b*x)^7,x,2,-cos(a+b*x)/b+cos(a+b*x)^3/b-3/5*cos(a+b*x)^5/b+1/7*cos(a+b*x)^7/b],
[sin(a+b*x)^8,x,5,35/128*x-35/128*cos(a+b*x)*sin(a+b*x)/b-35/192*cos(a+b*x)*sin(a+b*x)^3/b-7/48*cos(a+b*x)*sin(a+b*x)^5/b-1/8*cos(a+b*x)*sin(a+b*x)^7/b],

# Integrands of the form (b Sin[c+d x])^(n/2)
[sin(b*x)^(7/2),x,3,-10/21*sqrt(cos(1/4*Pi-1/2*b*x)^2)/cos(1/4*Pi-1/2*b*x)*EllipticF(sin(1/4*Pi-1/2*b*x),sqrt(2))/b-2/7*cos(b*x)*sin(b*x)^(5/2)/b-10/21*cos(b*x)*sqrt(sin(b*x))/b],
[sin(b*x)^(5/2),x,2,-6/5*sqrt(cos(1/4*Pi-1/2*b*x)^2)/cos(1/4*Pi-1/2*b*x)*EllipticE(sin(1/4*Pi-1/2*b*x),sqrt(2))/b-2/5*cos(b*x)*sin(b*x)^(3/2)/b],
[sin(b*x)^(3/2),x,2,-2/3*sqrt(cos(1/4*Pi-1/2*b*x)^2)/cos(1/4*Pi-1/2*b*x)*EllipticF(sin(1/4*Pi-1/2*b*x),sqrt(2))/b-2/3*cos(b*x)*sqrt(sin(b*x))/b],
[sin(b*x)^(1/2),x,1,-2*sqrt(cos(1/4*Pi-1/2*b*x)^2)/cos(1/4*Pi-1/2*b*x)*EllipticE(sin(1/4*Pi-1/2*b*x),sqrt(2))/b],
[1/sin(b*x)^(1/2),x,1,-2*sqrt(cos(1/4*Pi-1/2*b*x)^2)/cos(1/4*Pi-1/2*b*x)*EllipticF(sin(1/4*Pi-1/2*b*x),sqrt(2))/b],
[1/sin(b*x)^(3/2),x,2,2*sqrt(cos(1/4*Pi-1/2*b*x)^2)/cos(1/4*Pi-1/2*b*x)*EllipticE(sin(1/4*Pi-1/2*b*x),sqrt(2))/b-2*cos(b*x)/(b*sqrt(sin(b*x)))],
[1/sin(b*x)^(5/2),x,2,-2/3*sqrt(cos(1/4*Pi-1/2*b*x)^2)/cos(1/4*Pi-1/2*b*x)*EllipticF(sin(1/4*Pi-1/2*b*x),sqrt(2))/b-2/3*cos(b*x)/(b*sin(b*x)^(3/2))],
[1/sin(b*x)^(7/2),x,3,6/5*sqrt(cos(1/4*Pi-1/2*b*x)^2)/cos(1/4*Pi-1/2*b*x)*EllipticE(sin(1/4*Pi-1/2*b*x),sqrt(2))/b-2/5*cos(b*x)/(b*sin(b*x)^(5/2))-6/5*cos(b*x)/(b*sqrt(sin(b*x)))],
[sin(a+b*x)^(7/2),x,3,10/21*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticF(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))/b-2/7*cos(a+b*x)*sin(a+b*x)^(5/2)/b-10/21*cos(a+b*x)*sqrt(sin(a+b*x))/b],
[sin(a+b*x)^(5/2),x,2,6/5*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticE(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))/b-2/5*cos(a+b*x)*sin(a+b*x)^(3/2)/b],
[sin(a+b*x)^(3/2),x,2,2/3*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticF(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))/b-2/3*cos(a+b*x)*sqrt(sin(a+b*x))/b],
[sin(a+b*x)^(1/2),x,1,2*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticE(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))/b],
[1/sin(a+b*x)^(1/2),x,1,2*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticF(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))/b],
[1/sin(a+b*x)^(3/2),x,2,-2*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticE(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))/b-2*cos(a+b*x)/(b*sqrt(sin(a+b*x)))],
[1/sin(a+b*x)^(5/2),x,2,2/3*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticF(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))/b-2/3*cos(a+b*x)/(b*sin(a+b*x)^(3/2))],
[1/sin(a+b*x)^(7/2),x,3,-6/5*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticE(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))/b-2/5*cos(a+b*x)/(b*sin(a+b*x)^(5/2))-6/5*cos(a+b*x)/(b*sqrt(sin(a+b*x)))],
[(c*sin(a+b*x))^(7/2),x,4,-2/7*c*cos(a+b*x)*(c*sin(a+b*x))^(5/2)/b+10/21*c^4*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticF(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))*sqrt(sin(a+b*x))/(b*sqrt(c*sin(a+b*x)))-10/21*c^3*cos(a+b*x)*sqrt(c*sin(a+b*x))/b],
[(c*sin(a+b*x))^(5/2),x,3,-2/5*c*cos(a+b*x)*(c*sin(a+b*x))^(3/2)/b+6/5*c^2*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticE(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))*sqrt(c*sin(a+b*x))/(b*sqrt(sin(a+b*x)))],
[(c*sin(a+b*x))^(3/2),x,3,2/3*c^2*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticF(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))*sqrt(sin(a+b*x))/(b*sqrt(c*sin(a+b*x)))-2/3*c*cos(a+b*x)*sqrt(c*sin(a+b*x))/b],
[(c*sin(a+b*x))^(1/2),x,2,2*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticE(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))*sqrt(c*sin(a+b*x))/(b*sqrt(sin(a+b*x)))],
[1/(c*sin(a+b*x))^(1/2),x,2,2*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticF(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))*sqrt(sin(a+b*x))/(b*sqrt(c*sin(a+b*x)))],
[1/(c*sin(a+b*x))^(3/2),x,3,-2*cos(a+b*x)/(b*c*sqrt(c*sin(a+b*x)))-2*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticE(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))*sqrt(c*sin(a+b*x))/(b*c^2*sqrt(sin(a+b*x)))],
[1/(c*sin(a+b*x))^(5/2),x,3,-2/3*cos(a+b*x)/(b*c*(c*sin(a+b*x))^(3/2))+2/3*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticF(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))*sqrt(sin(a+b*x))/(b*c^2*sqrt(c*sin(a+b*x)))],
[1/(c*sin(a+b*x))^(7/2),x,4,-2/5*cos(a+b*x)/(b*c*(c*sin(a+b*x))^(5/2))-6/5*cos(a+b*x)/(b*c^3*sqrt(c*sin(a+b*x)))-6/5*sqrt(cos(1/2*(a-1/2*Pi+b*x))^2)/cos(1/2*(a-1/2*Pi+b*x))*EllipticE(sin(1/2*(a-1/2*Pi+b*x)),sqrt(2))*sqrt(c*sin(a+b*x))/(b*c^4*sqrt(sin(a+b*x)))],

# Integrands of the form (b Sin[c+d x])^(n/3)
[(c*sin(a+b*x))^(4/3),x,1,3/7*cos(a+b*x)*hypergeom([1/2,7/6],[13/6],sin(a+b*x)^2)*(c*sin(a+b*x))^(7/3)/(b*c*sqrt(cos(a+b*x)^2))],
[(c*sin(a+b*x))^(2/3),x,1,3/5*cos(a+b*x)*hypergeom([1/2,5/6],[11/6],sin(a+b*x)^2)*(c*sin(a+b*x))^(5/3)/(b*c*sqrt(cos(a+b*x)^2))],
[(c*sin(a+b*x))^(1/3),x,1,3/2*c^(1/3)*EllipticF(sqrt(2)*sqrt(1-(c*sin(a+b*x))^(2/3)/c^(2/3))/sqrt(3-I*sqrt(3)),sqrt((3*I+sqrt(3))/(3*I-sqrt(3))))*sec(a+b*x)*(1-I*sqrt(3))*sqrt(1-(c*sin(a+b*x))^(2/3)/c^(2/3))*sqrt((I-sqrt(3))/(3*I-sqrt(3))+2*(c*sin(a+b*x))^(2/3)/(c^(2/3)*(3+I*sqrt(3))))*sqrt(3-I*sqrt(3))*sqrt(2*(c*sin(a+b*x))^(2/3)/(c^(2/3)*(3-I*sqrt(3)))+(I+sqrt(3))/(3*I+sqrt(3)))/(b*sqrt(2))-3*c^(1/3)*EllipticE(sqrt(2)*sqrt(1-(c*sin(a+b*x))^(2/3)/c^(2/3))/sqrt(3+I*sqrt(3)),sqrt((3*I-sqrt(3))/(3*I+sqrt(3))))*sec(a+b*x)*sqrt(1-(c*sin(a+b*x))^(2/3)/c^(2/3))*sqrt((I-sqrt(3))/(3*I-sqrt(3))+2*(c*sin(a+b*x))^(2/3)/(c^(2/3)*(3+I*sqrt(3))))*sqrt(3/2*(3-I*sqrt(3)))*sqrt(2*(c*sin(a+b*x))^(2/3)/(c^(2/3)*(3-I*sqrt(3)))+(I+sqrt(3))/(3*I+sqrt(3)))/b,3/4*cos(a+b*x)*hypergeom([1/2,2/3],[5/3],sin(a+b*x)^2)*(c*sin(a+b*x))^(4/3)/(b*c*sqrt(cos(a+b*x)^2))],
[1/(c*sin(a+b*x))^(1/3),x,1,-3*EllipticF(sqrt(2)*sqrt(1-(c*sin(a+b*x))^(2/3)/c^(2/3))/sqrt(3-I*sqrt(3)),sqrt((3*I+sqrt(3))/(3*I-sqrt(3))))*sec(a+b*x)*sqrt(1-(c*sin(a+b*x))^(2/3)/c^(2/3))*sqrt((I-sqrt(3))/(3*I-sqrt(3))+2*(c*sin(a+b*x))^(2/3)/(c^(2/3)*(3+I*sqrt(3))))*sqrt(3-I*sqrt(3))*sqrt(2*(c*sin(a+b*x))^(2/3)/(c^(2/3)*(3-I*sqrt(3)))+(I+sqrt(3))/(3*I+sqrt(3)))/(b*c^(1/3)*sqrt(2)),3/2*cos(a+b*x)*hypergeom([1/3,1/2],[4/3],sin(a+b*x)^2)*(c*sin(a+b*x))^(2/3)/(b*c*sqrt(cos(a+b*x)^2))],
[1/(c*sin(a+b*x))^(2/3),x,1,1/2*3^(3/4)*sqrt(cos(arccos((c^(2/3)-(c*sin(a+b*x))^(2/3)*(1-sqrt(3)))/(c^(2/3)-(c*sin(a+b*x))^(2/3)*(1+sqrt(3)))))^2)/cos(arccos((c^(2/3)-(c*sin(a+b*x))^(2/3)*(1-sqrt(3)))/(c^(2/3)-(c*sin(a+b*x))^(2/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((c^(2/3)-(c*sin(a+b*x))^(2/3)*(1-sqrt(3)))/(c^(2/3)-(c*sin(a+b*x))^(2/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*sec(a+b*x)*(c*sin(a+b*x))^(1/3)*(c^(2/3)-(c*sin(a+b*x))^(2/3))*sqrt(c^(4/3)*(1+(c*sin(a+b*x))^(2/3)/c^(2/3)+(c*sin(a+b*x))^(4/3)/c^(4/3))/(c^(2/3)-(c*sin(a+b*x))^(2/3)*(1+sqrt(3)))^2)/(b*c^(5/3)*sqrt(-(c*sin(a+b*x))^(2/3)*(c^(2/3)-(c*sin(a+b*x))^(2/3))/(c^(2/3)-(c*sin(a+b*x))^(2/3)*(1+sqrt(3)))^2)),3*cos(a+b*x)*hypergeom([1/6,1/2],[7/6],sin(a+b*x)^2)*(c*sin(a+b*x))^(1/3)/(b*c*sqrt(cos(a+b*x)^2))],
[1/(c*sin(a+b*x))^(4/3),x,1,-3*cos(a+b*x)*hypergeom([-1/6,1/2],[5/6],sin(a+b*x)^2)/(b*c*(c*sin(a+b*x))^(1/3)*sqrt(cos(a+b*x)^2))],

# Integrands of the form (b Sin[c+d x])^n with n symbolic
[sin(a+b*x)^n,x,1,cos(a+b*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(a+b*x)^2)*sin(a+b*x)^(1+n)/(b*(1+n)*sqrt(cos(a+b*x)^2))],
[(c*sin(a+b*x))^n,x,1,cos(a+b*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+n)/(b*c*(1+n)*sqrt(cos(a+b*x)^2))],

# Integrands of the form (a Sin[e+f x])^m (b Trg[e+f x])^n

# Integrands of the form (a Sin[e+f x])^m (b Sin[e+f x])^n

# Integrands of the form (a Sin[e+f x])^m (b Sin[e+f x])^n with m and n symbolic
[(a*sin(e+f*x))^m*(b*sin(e+f*x))^n,x,2,cos(e+f*x)*hypergeom([1/2,1/2*(1+m+n)],[1/2*(3+m+n)],sin(e+f*x)^2)*(a*sin(e+f*x))^(1+m)*(b*sin(e+f*x))^n/(a*f*(1+m+n)*sqrt(cos(e+f*x)^2))],

# Integrands of the form (a Sin[e+f x])^m (b Cos[e+f x])^n

# Integrands of the form Sin[e+f x]^m Cos[e+f x]^n

# m>0
[cos(a+b*x)^3*sin(a+b*x),x,2,-1/4*cos(a+b*x)^4/b],
[cos(a+b*x)^2*sin(a+b*x),x,2,-1/3*cos(a+b*x)^3/b],
[cos(a+b*x)*sin(a+b*x),x,2,1/2*sin(a+b*x)^2/b],
[sec(a+b*x)*sin(a+b*x),x,1,-log(cos(a+b*x))/b],
[sec(a+b*x)^2*sin(a+b*x),x,2,sec(a+b*x)/b],
[sec(a+b*x)^3*sin(a+b*x),x,2,1/2*sec(a+b*x)^2/b],
[sec(a+b*x)^4*sin(a+b*x),x,2,1/3*sec(a+b*x)^3/b],
[cos(a+b*x)^7*sin(a+b*x)^2,x,3,1/3*sin(a+b*x)^3/b-3/5*sin(a+b*x)^5/b+3/7*sin(a+b*x)^7/b-1/9*sin(a+b*x)^9/b],
[cos(a+b*x)^5*sin(a+b*x)^2,x,3,1/3*sin(a+b*x)^3/b-2/5*sin(a+b*x)^5/b+1/7*sin(a+b*x)^7/b],
[cos(a+b*x)^3*sin(a+b*x)^2,x,3,1/3*sin(a+b*x)^3/b-1/5*sin(a+b*x)^5/b],
[cos(a+b*x)*sin(a+b*x)^2,x,2,1/3*sin(a+b*x)^3/b],
[sec(a+b*x)^2*sin(a+b*x)^2,x,2,-x+tan(a+b*x)/b],
[sec(a+b*x)^4*sin(a+b*x)^2,x,2,1/3*tan(a+b*x)^3/b],
[sec(a+b*x)^6*sin(a+b*x)^2,x,3,1/3*tan(a+b*x)^3/b+1/5*tan(a+b*x)^5/b],
[sec(a+b*x)^8*sin(a+b*x)^2,x,3,1/3*tan(a+b*x)^3/b+2/5*tan(a+b*x)^5/b+1/7*tan(a+b*x)^7/b],
[sec(a+b*x)^10*sin(a+b*x)^2,x,3,1/3*tan(a+b*x)^3/b+3/5*tan(a+b*x)^5/b+3/7*tan(a+b*x)^7/b+1/9*tan(a+b*x)^9/b],
[cos(a+b*x)^6*sin(a+b*x)^2,x,5,5/128*x+5/128*cos(a+b*x)*sin(a+b*x)/b+5/192*cos(a+b*x)^3*sin(a+b*x)/b+1/48*cos(a+b*x)^5*sin(a+b*x)/b-1/8*cos(a+b*x)^7*sin(a+b*x)/b],
[cos(a+b*x)^4*sin(a+b*x)^2,x,4,1/16*x+1/16*cos(a+b*x)*sin(a+b*x)/b+1/24*cos(a+b*x)^3*sin(a+b*x)/b-1/6*cos(a+b*x)^5*sin(a+b*x)/b],
[cos(a+b*x)^2*sin(a+b*x)^2,x,3,1/8*x+1/8*cos(a+b*x)*sin(a+b*x)/b-1/4*cos(a+b*x)^3*sin(a+b*x)/b],
[sin(a+b*x)^2,x,2,1/2*x-1/2*cos(a+b*x)*sin(a+b*x)/b],
[sec(a+b*x)*sin(a+b*x)^2,x,3,arctanh(sin(a+b*x))/b-sin(a+b*x)/b],
[sec(a+b*x)^3*sin(a+b*x)^2,x,2,-1/2*arctanh(sin(a+b*x))/b+1/2*sec(a+b*x)*tan(a+b*x)/b],
[sec(a+b*x)^5*sin(a+b*x)^2,x,3,-1/8*arctanh(sin(a+b*x))/b-1/8*sec(a+b*x)*tan(a+b*x)/b+1/4*sec(a+b*x)^3*tan(a+b*x)/b],
[sec(a+b*x)^7*sin(a+b*x)^2,x,4,-1/16*arctanh(sin(a+b*x))/b-1/16*sec(a+b*x)*tan(a+b*x)/b-1/24*sec(a+b*x)^3*tan(a+b*x)/b+1/6*sec(a+b*x)^5*tan(a+b*x)/b],
[cos(a+b*x)^5*sin(a+b*x)^3,x,3,-1/6*cos(a+b*x)^6/b+1/8*cos(a+b*x)^8/b],
[cos(a+b*x)^4*sin(a+b*x)^3,x,3,-1/5*cos(a+b*x)^5/b+1/7*cos(a+b*x)^7/b],
[cos(a+b*x)^3*sin(a+b*x)^3,x,3,1/4*sin(a+b*x)^4/b-1/6*sin(a+b*x)^6/b],
[cos(a+b*x)^2*sin(a+b*x)^3,x,3,-1/3*cos(a+b*x)^3/b+1/5*cos(a+b*x)^5/b],
[cos(a+b*x)*sin(a+b*x)^3,x,2,1/4*sin(a+b*x)^4/b],
[sec(a+b*x)*sin(a+b*x)^3,x,3,1/2*cos(a+b*x)^2/b-log(cos(a+b*x))/b],
[sec(a+b*x)^2*sin(a+b*x)^3,x,3,cos(a+b*x)/b+sec(a+b*x)/b],
[sec(a+b*x)^3*sin(a+b*x)^3,x,2,log(cos(a+b*x))/b+1/2*tan(a+b*x)^2/b],
[sec(a+b*x)^4*sin(a+b*x)^3,x,2,-sec(a+b*x)/b+1/3*sec(a+b*x)^3/b],
[sec(a+b*x)^5*sin(a+b*x)^3,x,2,1/4*tan(a+b*x)^4/b],
[sec(a+b*x)^6*sin(a+b*x)^3,x,3,-1/3*sec(a+b*x)^3/b+1/5*sec(a+b*x)^5/b],
[sec(a+b*x)^7*sin(a+b*x)^3,x,3,-1/4*sec(a+b*x)^4/b+1/6*sec(a+b*x)^6/b],
[sec(a+b*x)^8*sin(a+b*x)^3,x,3,-1/5*sec(a+b*x)^5/b+1/7*sec(a+b*x)^7/b],
[sec(a+b*x)^9*sin(a+b*x)^3,x,3,-1/6*sec(a+b*x)^6/b+1/8*sec(a+b*x)^8/b],
[cos(a+b*x)^7*sin(a+b*x)^4,x,3,1/5*sin(a+b*x)^5/b-3/7*sin(a+b*x)^7/b+1/3*sin(a+b*x)^9/b-1/11*sin(a+b*x)^11/b],
[cos(a+b*x)^5*sin(a+b*x)^4,x,3,1/5*sin(a+b*x)^5/b-2/7*sin(a+b*x)^7/b+1/9*sin(a+b*x)^9/b],
[cos(a+b*x)^3*sin(a+b*x)^4,x,3,1/5*sin(a+b*x)^5/b-1/7*sin(a+b*x)^7/b],
[cos(a+b*x)*sin(a+b*x)^4,x,2,1/5*sin(a+b*x)^5/b],
[sec(a+b*x)^2*sin(a+b*x)^4,x,4,-3/2*x+3/2*tan(a+b*x)/b-1/2*sin(a+b*x)^2*tan(a+b*x)/b],
[sec(a+b*x)^4*sin(a+b*x)^4,x,3,x-tan(a+b*x)/b+1/3*tan(a+b*x)^3/b],
[sec(a+b*x)^6*sin(a+b*x)^4,x,2,1/5*tan(a+b*x)^5/b],
[sec(a+b*x)^8*sin(a+b*x)^4,x,3,1/5*tan(a+b*x)^5/b+1/7*tan(a+b*x)^7/b],
[sec(a+b*x)^10*sin(a+b*x)^4,x,3,1/5*tan(a+b*x)^5/b+2/7*tan(a+b*x)^7/b+1/9*tan(a+b*x)^9/b],
[cos(a+b*x)^6*sin(a+b*x)^4,x,6,3/256*x+3/256*cos(a+b*x)*sin(a+b*x)/b+1/128*cos(a+b*x)^3*sin(a+b*x)/b+1/160*cos(a+b*x)^5*sin(a+b*x)/b-3/80*cos(a+b*x)^7*sin(a+b*x)/b-1/10*cos(a+b*x)^7*sin(a+b*x)^3/b],
[cos(a+b*x)^4*sin(a+b*x)^4,x,5,3/128*x+3/128*cos(a+b*x)*sin(a+b*x)/b+1/64*cos(a+b*x)^3*sin(a+b*x)/b-1/16*cos(a+b*x)^5*sin(a+b*x)/b-1/8*cos(a+b*x)^5*sin(a+b*x)^3/b],
[cos(a+b*x)^2*sin(a+b*x)^4,x,4,1/16*x+1/16*cos(a+b*x)*sin(a+b*x)/b-1/8*cos(a+b*x)^3*sin(a+b*x)/b-1/6*cos(a+b*x)^3*sin(a+b*x)^3/b],
[sin(a+b*x)^4,x,3,3/8*x-3/8*cos(a+b*x)*sin(a+b*x)/b-1/4*cos(a+b*x)*sin(a+b*x)^3/b],
[sec(a+b*x)*sin(a+b*x)^4,x,4,arctanh(sin(a+b*x))/b-sin(a+b*x)/b-1/3*sin(a+b*x)^3/b],
[sec(a+b*x)^3*sin(a+b*x)^4,x,4,-3/2*arctanh(sin(a+b*x))/b+3/2*sin(a+b*x)/b+1/2*sin(a+b*x)*tan(a+b*x)^2/b],
[sec(a+b*x)^5*sin(a+b*x)^4,x,3,3/8*arctanh(sin(a+b*x))/b-3/8*sec(a+b*x)*tan(a+b*x)/b+1/4*sec(a+b*x)*tan(a+b*x)^3/b],
[sec(a+b*x)^7*sin(a+b*x)^4,x,4,1/16*arctanh(sin(a+b*x))/b+1/16*sec(a+b*x)*tan(a+b*x)/b-1/8*sec(a+b*x)^3*tan(a+b*x)/b+1/6*sec(a+b*x)^3*tan(a+b*x)^3/b],
[sec(a+b*x)^9*sin(a+b*x)^4,x,5,3/128*arctanh(sin(a+b*x))/b+3/128*sec(a+b*x)*tan(a+b*x)/b+1/64*sec(a+b*x)^3*tan(a+b*x)/b-1/16*sec(a+b*x)^5*tan(a+b*x)/b+1/8*sec(a+b*x)^5*tan(a+b*x)^3/b],
[cos(a+b*x)^7*sin(a+b*x)^5,x,4,-1/8*cos(a+b*x)^8/b+1/5*cos(a+b*x)^10/b-1/12*cos(a+b*x)^12/b],
[cos(a+b*x)^6*sin(a+b*x)^5,x,3,-1/7*cos(a+b*x)^7/b+2/9*cos(a+b*x)^9/b-1/11*cos(a+b*x)^11/b],
[cos(a+b*x)^5*sin(a+b*x)^5,x,4,1/6*sin(a+b*x)^6/b-1/4*sin(a+b*x)^8/b+1/10*sin(a+b*x)^10/b],
[cos(a+b*x)^4*sin(a+b*x)^5,x,3,-1/5*cos(a+b*x)^5/b+2/7*cos(a+b*x)^7/b-1/9*cos(a+b*x)^9/b],
[cos(a+b*x)^3*sin(a+b*x)^5,x,3,1/6*sin(a+b*x)^6/b-1/8*sin(a+b*x)^8/b],
[cos(a+b*x)^2*sin(a+b*x)^5,x,3,-1/3*cos(a+b*x)^3/b+2/5*cos(a+b*x)^5/b-1/7*cos(a+b*x)^7/b],
[cos(a+b*x)*sin(a+b*x)^5,x,2,1/6*sin(a+b*x)^6/b],
[sec(a+b*x)*sin(a+b*x)^5,x,4,cos(a+b*x)^2/b-1/4*cos(a+b*x)^4/b-log(cos(a+b*x))/b],
[sec(a+b*x)^2*sin(a+b*x)^5,x,3,2*cos(a+b*x)/b-1/3*cos(a+b*x)^3/b+sec(a+b*x)/b],
[sec(a+b*x)^3*sin(a+b*x)^5,x,4,-1/2*cos(a+b*x)^2/b+2*log(cos(a+b*x))/b+1/2*sec(a+b*x)^2/b],
[sec(a+b*x)^4*sin(a+b*x)^5,x,3,-cos(a+b*x)/b-2*sec(a+b*x)/b+1/3*sec(a+b*x)^3/b],
[sec(a+b*x)^5*sin(a+b*x)^5,x,3,-log(cos(a+b*x))/b-1/2*tan(a+b*x)^2/b+1/4*tan(a+b*x)^4/b],
[sec(a+b*x)^6*sin(a+b*x)^5,x,3,sec(a+b*x)/b-2/3*sec(a+b*x)^3/b+1/5*sec(a+b*x)^5/b],
[sec(a+b*x)^7*sin(a+b*x)^5,x,2,1/6*tan(a+b*x)^6/b],
[sec(a+b*x)^8*sin(a+b*x)^5,x,3,1/3*sec(a+b*x)^3/b-2/5*sec(a+b*x)^5/b+1/7*sec(a+b*x)^7/b],
[sec(a+b*x)^9*sin(a+b*x)^5,x,3,1/6*tan(a+b*x)^6/b+1/8*tan(a+b*x)^8/b],
[sec(a+b*x)^10*sin(a+b*x)^5,x,3,1/5*sec(a+b*x)^5/b-2/7*sec(a+b*x)^7/b+1/9*sec(a+b*x)^9/b],
[sec(a+b*x)^11*sin(a+b*x)^5,x,4,1/6*sec(a+b*x)^6/b-1/4*sec(a+b*x)^8/b+1/10*sec(a+b*x)^10/b],
[sec(a+b*x)^12*sin(a+b*x)^5,x,3,1/7*sec(a+b*x)^7/b-2/9*sec(a+b*x)^9/b+1/11*sec(a+b*x)^11/b],
[sec(a+b*x)^13*sin(a+b*x)^5,x,4,1/8*sec(a+b*x)^8/b-1/5*sec(a+b*x)^10/b+1/12*sec(a+b*x)^12/b],
[sec(a+b*x)^3*sin(a+b*x)^6,x,5,-5/2*arctanh(sin(a+b*x))/b+5/2*sin(a+b*x)/b+5/6*sin(a+b*x)^3/b+1/2*sin(a+b*x)^3*tan(a+b*x)^2/b],
[sec(a+b*x)^6*sin(a+b*x)^7,x,3,cos(a+b*x)/b+3*sec(a+b*x)/b-sec(a+b*x)^3/b+1/5*sec(a+b*x)^5/b],

# m<0
[cos(a+b*x)^6/sin(a+b*x),x,4,-arctanh(cos(a+b*x))/b+cos(a+b*x)/b+1/3*cos(a+b*x)^3/b+1/5*cos(a+b*x)^5/b],
[cos(a+b*x)^5/sin(a+b*x),x,4,log(sin(a+b*x))/b-sin(a+b*x)^2/b+1/4*sin(a+b*x)^4/b],
[cos(a+b*x)^4/sin(a+b*x),x,4,-arctanh(cos(a+b*x))/b+cos(a+b*x)/b+1/3*cos(a+b*x)^3/b],
[cos(a+b*x)^3/sin(a+b*x),x,3,log(sin(a+b*x))/b-1/2*sin(a+b*x)^2/b],
[cos(a+b*x)^2/sin(a+b*x),x,3,-arctanh(cos(a+b*x))/b+cos(a+b*x)/b],
[cos(a+b*x)/sin(a+b*x),x,1,log(sin(a+b*x))/b],
[sec(a+b*x)/sin(a+b*x),x,2,log(tan(a+b*x))/b],
[sec(a+b*x)^2/sin(a+b*x),x,3,-arctanh(cos(a+b*x))/b+sec(a+b*x)/b],
[sec(a+b*x)^3/sin(a+b*x),x,3,log(tan(a+b*x))/b+1/2*tan(a+b*x)^2/b],
[sec(a+b*x)^4/sin(a+b*x),x,4,-arctanh(cos(a+b*x))/b+sec(a+b*x)/b+1/3*sec(a+b*x)^3/b],
[sec(a+b*x)^5/sin(a+b*x),x,4,log(tan(a+b*x))/b+tan(a+b*x)^2/b+1/4*tan(a+b*x)^4/b],
[sec(a+b*x)^6/sin(a+b*x),x,4,-arctanh(cos(a+b*x))/b+sec(a+b*x)/b+1/3*sec(a+b*x)^3/b+1/5*sec(a+b*x)^5/b],
[sec(a+b*x)^7/sin(a+b*x),x,4,log(tan(a+b*x))/b+3/2*tan(a+b*x)^2/b+3/4*tan(a+b*x)^4/b+1/6*tan(a+b*x)^6/b],
[cos(a+b*x)^7/sin(a+b*x)^2,x,3,-csc(a+b*x)/b-3*sin(a+b*x)/b+sin(a+b*x)^3/b-1/5*sin(a+b*x)^5/b],
[cos(a+b*x)^6/sin(a+b*x)^2,x,5,-15/8*x-15/8*cot(a+b*x)/b+5/8*cos(a+b*x)^2*cot(a+b*x)/b+1/4*cos(a+b*x)^4*cot(a+b*x)/b],
[cos(a+b*x)^5/sin(a+b*x)^2,x,3,-csc(a+b*x)/b-2*sin(a+b*x)/b+1/3*sin(a+b*x)^3/b],
[cos(a+b*x)^4/sin(a+b*x)^2,x,4,-3/2*x-3/2*cot(a+b*x)/b+1/2*cos(a+b*x)^2*cot(a+b*x)/b],
[cos(a+b*x)^3/sin(a+b*x)^2,x,3,-csc(a+b*x)/b-sin(a+b*x)/b],
[cos(a+b*x)^2/sin(a+b*x)^2,x,2,-x-cot(a+b*x)/b],
[cos(a+b*x)/sin(a+b*x)^2,x,2,-csc(a+b*x)/b],
[sec(a+b*x)/sin(a+b*x)^2,x,3,arctanh(sin(a+b*x))/b-csc(a+b*x)/b],
[sec(a+b*x)^2/sin(a+b*x)^2,x,3,-cot(a+b*x)/b+tan(a+b*x)/b],
[sec(a+b*x)^3/sin(a+b*x)^2,x,4,3/2*arctanh(sin(a+b*x))/b-3/2*csc(a+b*x)/b+1/2*csc(a+b*x)*sec(a+b*x)^2/b],
[sec(a+b*x)^4/sin(a+b*x)^2,x,3,-cot(a+b*x)/b+2*tan(a+b*x)/b+1/3*tan(a+b*x)^3/b],
[sec(a+b*x)^5/sin(a+b*x)^2,x,5,15/8*arctanh(sin(a+b*x))/b-15/8*csc(a+b*x)/b+5/8*csc(a+b*x)*sec(a+b*x)^2/b+1/4*csc(a+b*x)*sec(a+b*x)^4/b],
[cos(a+b*x)^7/sin(a+b*x)^3,x,4,-1/2*csc(a+b*x)^2/b-3*log(sin(a+b*x))/b+3/2*sin(a+b*x)^2/b-1/4*sin(a+b*x)^4/b],
[cos(a+b*x)^6/sin(a+b*x)^3,x,5,5/2*arctanh(cos(a+b*x))/b-5/2*cos(a+b*x)/b-5/6*cos(a+b*x)^3/b-1/2*cos(a+b*x)^3*cot(a+b*x)^2/b],
[cos(a+b*x)^5/sin(a+b*x)^3,x,4,-1/2*csc(a+b*x)^2/b-2*log(sin(a+b*x))/b+1/2*sin(a+b*x)^2/b],
[cos(a+b*x)^4/sin(a+b*x)^3,x,4,3/2*arctanh(cos(a+b*x))/b-3/2*cos(a+b*x)/b-1/2*cos(a+b*x)*cot(a+b*x)^2/b],
[cos(a+b*x)^3/sin(a+b*x)^3,x,2,-1/2*cot(a+b*x)^2/b-log(sin(a+b*x))/b],
[cos(a+b*x)^2/sin(a+b*x)^3,x,2,1/2*arctanh(cos(a+b*x))/b-1/2*cot(a+b*x)*csc(a+b*x)/b],
[cos(a+b*x)/sin(a+b*x)^3,x,2,-1/2*csc(a+b*x)^2/b],
[sec(a+b*x)/sin(a+b*x)^3,x,3,-1/2*cot(a+b*x)^2/b+log(tan(a+b*x))/b],
[sec(a+b*x)^2/sin(a+b*x)^3,x,4,-3/2*arctanh(cos(a+b*x))/b+3/2*sec(a+b*x)/b-1/2*csc(a+b*x)^2*sec(a+b*x)/b],
[sec(a+b*x)^3/sin(a+b*x)^3,x,4,-1/2*cot(a+b*x)^2/b+2*log(tan(a+b*x))/b+1/2*tan(a+b*x)^2/b],
[sec(a+b*x)^4/sin(a+b*x)^3,x,5,-5/2*arctanh(cos(a+b*x))/b+5/2*sec(a+b*x)/b+5/6*sec(a+b*x)^3/b-1/2*csc(a+b*x)^2*sec(a+b*x)^3/b],
[sec(a+b*x)^5/sin(a+b*x)^3,x,4,-1/2*cot(a+b*x)^2/b+3*log(tan(a+b*x))/b+3/2*tan(a+b*x)^2/b+1/4*tan(a+b*x)^4/b],
[cos(a+b*x)^9/sin(a+b*x)^4,x,3,4*csc(a+b*x)/b-1/3*csc(a+b*x)^3/b+6*sin(a+b*x)/b-4/3*sin(a+b*x)^3/b+1/5*sin(a+b*x)^5/b],
[cos(a+b*x)^8/sin(a+b*x)^4,x,6,35/8*x+35/8*cot(a+b*x)/b-35/24*cot(a+b*x)^3/b+7/8*cos(a+b*x)^2*cot(a+b*x)^3/b+1/4*cos(a+b*x)^4*cot(a+b*x)^3/b],
[cos(a+b*x)^7/sin(a+b*x)^4,x,3,3*csc(a+b*x)/b-1/3*csc(a+b*x)^3/b+3*sin(a+b*x)/b-1/3*sin(a+b*x)^3/b],
[cos(a+b*x)^6/sin(a+b*x)^4,x,5,5/2*x+5/2*cot(a+b*x)/b-5/6*cot(a+b*x)^3/b+1/2*cos(a+b*x)^2*cot(a+b*x)^3/b],
[cos(a+b*x)^5/sin(a+b*x)^4,x,3,2*csc(a+b*x)/b-1/3*csc(a+b*x)^3/b+sin(a+b*x)/b],
[cos(a+b*x)^4/sin(a+b*x)^4,x,3,x+cot(a+b*x)/b-1/3*cot(a+b*x)^3/b],
[cos(a+b*x)^3/sin(a+b*x)^4,x,2,csc(a+b*x)/b-1/3*csc(a+b*x)^3/b],
[cos(a+b*x)^2/sin(a+b*x)^4,x,2,-1/3*cot(a+b*x)^3/b],
[cos(a+b*x)/sin(a+b*x)^4,x,2,-1/3*csc(a+b*x)^3/b],
[sec(a+b*x)/sin(a+b*x)^4,x,4,arctanh(sin(a+b*x))/b-csc(a+b*x)/b-1/3*csc(a+b*x)^3/b],
[sec(a+b*x)^2/sin(a+b*x)^4,x,3,-2*cot(a+b*x)/b-1/3*cot(a+b*x)^3/b+tan(a+b*x)/b],
[sec(a+b*x)^3/sin(a+b*x)^4,x,5,5/2*arctanh(sin(a+b*x))/b-5/2*csc(a+b*x)/b-5/6*csc(a+b*x)^3/b+1/2*csc(a+b*x)^3*sec(a+b*x)^2/b],
[sec(a+b*x)^4/sin(a+b*x)^4,x,3,-3*cot(a+b*x)/b-1/3*cot(a+b*x)^3/b+3*tan(a+b*x)/b+1/3*tan(a+b*x)^3/b],
[sec(a+b*x)^5/sin(a+b*x)^4,x,6,35/8*arctanh(sin(a+b*x))/b-35/8*csc(a+b*x)/b-35/24*csc(a+b*x)^3/b+7/8*csc(a+b*x)^3*sec(a+b*x)^2/b+1/4*csc(a+b*x)^3*sec(a+b*x)^4/b],
[cos(a+b*x)^9/sin(a+b*x)^5,x,4,2*csc(a+b*x)^2/b-1/4*csc(a+b*x)^4/b+6*log(sin(a+b*x))/b-2*sin(a+b*x)^2/b+1/4*sin(a+b*x)^4/b],
[cos(a+b*x)^8/sin(a+b*x)^5,x,6,-35/8*arctanh(cos(a+b*x))/b+35/8*cos(a+b*x)/b+35/24*cos(a+b*x)^3/b+7/8*cos(a+b*x)^3*cot(a+b*x)^2/b-1/4*cos(a+b*x)^3*cot(a+b*x)^4/b],
[cos(a+b*x)^7/sin(a+b*x)^5,x,4,3/2*csc(a+b*x)^2/b-1/4*csc(a+b*x)^4/b+3*log(sin(a+b*x))/b-1/2*sin(a+b*x)^2/b],
[cos(a+b*x)^6/sin(a+b*x)^5,x,5,-15/8*arctanh(cos(a+b*x))/b+15/8*cos(a+b*x)/b+5/8*cos(a+b*x)*cot(a+b*x)^2/b-1/4*cos(a+b*x)*cot(a+b*x)^4/b],
[cos(a+b*x)^5/sin(a+b*x)^5,x,3,1/2*cot(a+b*x)^2/b-1/4*cot(a+b*x)^4/b+log(sin(a+b*x))/b],
[cos(a+b*x)^4/sin(a+b*x)^5,x,3,-3/8*arctanh(cos(a+b*x))/b+3/8*cot(a+b*x)*csc(a+b*x)/b-1/4*cot(a+b*x)^3*csc(a+b*x)/b],
[cos(a+b*x)^3/sin(a+b*x)^5,x,2,-1/4*cot(a+b*x)^4/b],
[cos(a+b*x)^2/sin(a+b*x)^5,x,3,1/8*arctanh(cos(a+b*x))/b+1/8*cot(a+b*x)*csc(a+b*x)/b-1/4*cot(a+b*x)*csc(a+b*x)^3/b],
[cos(a+b*x)/sin(a+b*x)^5,x,2,-1/4*csc(a+b*x)^4/b],
[sec(a+b*x)/sin(a+b*x)^5,x,4,-cot(a+b*x)^2/b-1/4*cot(a+b*x)^4/b+log(tan(a+b*x))/b],
[sec(a+b*x)^2/sin(a+b*x)^5,x,5,-15/8*arctanh(cos(a+b*x))/b+15/8*sec(a+b*x)/b-5/8*csc(a+b*x)^2*sec(a+b*x)/b-1/4*csc(a+b*x)^4*sec(a+b*x)/b],
[sec(a+b*x)^3/sin(a+b*x)^5,x,4,-3/2*cot(a+b*x)^2/b-1/4*cot(a+b*x)^4/b+3*log(tan(a+b*x))/b+1/2*tan(a+b*x)^2/b],
[sec(a+b*x)^4/sin(a+b*x)^5,x,6,-35/8*arctanh(cos(a+b*x))/b+35/8*sec(a+b*x)/b+35/24*sec(a+b*x)^3/b-7/8*csc(a+b*x)^2*sec(a+b*x)^3/b-1/4*csc(a+b*x)^4*sec(a+b*x)^3/b],
[sec(a+b*x)^5/sin(a+b*x)^5,x,4,-2*cot(a+b*x)^2/b-1/4*cot(a+b*x)^4/b+6*log(tan(a+b*x))/b+2*tan(a+b*x)^2/b+1/4*tan(a+b*x)^4/b],
[cos(x)^2/sin(x)^6,x,3,-1/3*cot(x)^3-1/5*cot(x)^5],
[cos(x)^3/sin(x)^7,x,3,1/4*csc(x)^4-1/6*csc(x)^6],

# Integrands of the form Sin[e+f x]^m (b Cos[e+f x])^(n/2)

# m>0
[(d*cos(a+b*x))^(3/2)*sin(a+b*x),x,2,-2/5*(d*cos(a+b*x))^(5/2)/(b*d)],
[(d*cos(a+b*x))^(1/2)*sin(a+b*x),x,2,-2/3*(d*cos(a+b*x))^(3/2)/(b*d)],
[sin(a+b*x)/(d*cos(a+b*x))^(1/2),x,2,-2*sqrt(d*cos(a+b*x))/(b*d)],
[sin(a+b*x)/(d*cos(a+b*x))^(3/2),x,2,2/(b*d*sqrt(d*cos(a+b*x)))],
[sin(a+b*x)/(d*cos(a+b*x))^(5/2),x,2,2/3/(b*d*(d*cos(a+b*x))^(3/2))],
[sin(a+b*x)/(d*cos(a+b*x))^(7/2),x,2,2/5/(b*d*(d*cos(a+b*x))^(5/2))],
[sin(a+b*x)/(d*cos(a+b*x))^(9/2),x,2,2/7/(b*d*(d*cos(a+b*x))^(7/2))],
[(d*cos(a+b*x))^(9/2)*sin(a+b*x)^2,x,5,28/585*d^3*(d*cos(a+b*x))^(3/2)*sin(a+b*x)/b+4/117*d*(d*cos(a+b*x))^(7/2)*sin(a+b*x)/b-2/13*(d*cos(a+b*x))^(11/2)*sin(a+b*x)/(b*d)+28/195*d^4*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[(d*cos(a+b*x))^(7/2)*sin(a+b*x)^2,x,5,4/77*d*(d*cos(a+b*x))^(5/2)*sin(a+b*x)/b-2/11*(d*cos(a+b*x))^(9/2)*sin(a+b*x)/(b*d)+20/231*d^4*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))+20/231*d^3*sin(a+b*x)*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(5/2)*sin(a+b*x)^2,x,4,4/45*d*(d*cos(a+b*x))^(3/2)*sin(a+b*x)/b-2/9*(d*cos(a+b*x))^(7/2)*sin(a+b*x)/(b*d)+4/15*d^2*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[(d*cos(a+b*x))^(3/2)*sin(a+b*x)^2,x,4,-2/7*(d*cos(a+b*x))^(5/2)*sin(a+b*x)/(b*d)+4/21*d^2*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))+4/21*d*sin(a+b*x)*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(1/2)*sin(a+b*x)^2,x,3,-2/5*(d*cos(a+b*x))^(3/2)*sin(a+b*x)/(b*d)+4/5*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[sin(a+b*x)^2/(d*cos(a+b*x))^(1/2),x,3,4/3*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))-2/3*sin(a+b*x)*sqrt(d*cos(a+b*x))/(b*d)],
[sin(a+b*x)^2/(d*cos(a+b*x))^(3/2),x,3,2*sin(a+b*x)/(b*d*sqrt(d*cos(a+b*x)))-4*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*d^2*sqrt(cos(a+b*x)))],
[sin(a+b*x)^2/(d*cos(a+b*x))^(5/2),x,3,2/3*sin(a+b*x)/(b*d*(d*cos(a+b*x))^(3/2))-4/3*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*d^2*sqrt(d*cos(a+b*x)))],
[sin(a+b*x)^2/(d*cos(a+b*x))^(7/2),x,4,2/5*sin(a+b*x)/(b*d*(d*cos(a+b*x))^(5/2))-4/5*sin(a+b*x)/(b*d^3*sqrt(d*cos(a+b*x)))+4/5*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*d^4*sqrt(cos(a+b*x)))],
[sin(a+b*x)^2/(d*cos(a+b*x))^(9/2),x,4,2/7*sin(a+b*x)/(b*d*(d*cos(a+b*x))^(7/2))-4/21*sin(a+b*x)/(b*d^3*(d*cos(a+b*x))^(3/2))-4/21*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*d^4*sqrt(d*cos(a+b*x)))],
[(d*cos(a+b*x))^(1/2)*sin(a+b*x)^3,x,3,-2/3*(d*cos(a+b*x))^(3/2)/(b*d)+2/7*(d*cos(a+b*x))^(7/2)/(b*d^3)],
[sin(a+b*x)^3/(d*cos(a+b*x))^(1/2),x,3,2/5*(d*cos(a+b*x))^(5/2)/(b*d^3)-2*sqrt(d*cos(a+b*x))/(b*d)],
[sin(a+b*x)^3/(d*cos(a+b*x))^(3/2),x,3,2/3*(d*cos(a+b*x))^(3/2)/(b*d^3)+2/(b*d*sqrt(d*cos(a+b*x)))],
[sin(a+b*x)^3/(d*cos(a+b*x))^(5/2),x,3,2/3/(b*d*(d*cos(a+b*x))^(3/2))+2*sqrt(d*cos(a+b*x))/(b*d^3)],
[sin(a+b*x)^3/(d*cos(a+b*x))^(7/2),x,3,2/5/(b*d*(d*cos(a+b*x))^(5/2))+(-2)/(b*d^3*sqrt(d*cos(a+b*x)))],
[sin(a+b*x)^3/(d*cos(a+b*x))^(9/2),x,3,2/7/(b*d*(d*cos(a+b*x))^(7/2))+(-2/3)/(b*d^3*(d*cos(a+b*x))^(3/2))],
[sin(a+b*x)^3/(d*cos(a+b*x))^(11/2),x,3,2/9/(b*d*(d*cos(a+b*x))^(9/2))+(-2/5)/(b*d^3*(d*cos(a+b*x))^(5/2))],
[(d*cos(a+b*x))^(9/2)*sin(a+b*x)^4,x,6,56/3315*d^3*(d*cos(a+b*x))^(3/2)*sin(a+b*x)/b+8/663*d*(d*cos(a+b*x))^(7/2)*sin(a+b*x)/b-12/221*(d*cos(a+b*x))^(11/2)*sin(a+b*x)/(b*d)-2/17*(d*cos(a+b*x))^(11/2)*sin(a+b*x)^3/(b*d)+56/1105*d^4*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[(d*cos(a+b*x))^(7/2)*sin(a+b*x)^4,x,6,8/385*d*(d*cos(a+b*x))^(5/2)*sin(a+b*x)/b-4/55*(d*cos(a+b*x))^(9/2)*sin(a+b*x)/(b*d)-2/15*(d*cos(a+b*x))^(9/2)*sin(a+b*x)^3/(b*d)+8/231*d^4*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))+8/231*d^3*sin(a+b*x)*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(5/2)*sin(a+b*x)^4,x,5,8/195*d*(d*cos(a+b*x))^(3/2)*sin(a+b*x)/b-4/39*(d*cos(a+b*x))^(7/2)*sin(a+b*x)/(b*d)-2/13*(d*cos(a+b*x))^(7/2)*sin(a+b*x)^3/(b*d)+8/65*d^2*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[(d*cos(a+b*x))^(3/2)*sin(a+b*x)^4,x,5,-12/77*(d*cos(a+b*x))^(5/2)*sin(a+b*x)/(b*d)-2/11*(d*cos(a+b*x))^(5/2)*sin(a+b*x)^3/(b*d)+8/77*d^2*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))+8/77*d*sin(a+b*x)*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(1/2)*sin(a+b*x)^4,x,4,-4/15*(d*cos(a+b*x))^(3/2)*sin(a+b*x)/(b*d)-2/9*(d*cos(a+b*x))^(3/2)*sin(a+b*x)^3/(b*d)+8/15*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[sin(a+b*x)^4/(d*cos(a+b*x))^(1/2),x,4,8/7*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))-4/7*sin(a+b*x)*sqrt(d*cos(a+b*x))/(b*d)-2/7*sin(a+b*x)^3*sqrt(d*cos(a+b*x))/(b*d)],
[sin(a+b*x)^4/(d*cos(a+b*x))^(3/2),x,4,12/5*(d*cos(a+b*x))^(3/2)*sin(a+b*x)/(b*d^3)+2*sin(a+b*x)^3/(b*d*sqrt(d*cos(a+b*x)))-24/5*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*d^2*sqrt(cos(a+b*x)))],
[sin(a+b*x)^4/(d*cos(a+b*x))^(5/2),x,4,2/3*sin(a+b*x)^3/(b*d*(d*cos(a+b*x))^(3/2))-8/3*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*d^2*sqrt(d*cos(a+b*x)))+4/3*sin(a+b*x)*sqrt(d*cos(a+b*x))/(b*d^3)],
[sin(a+b*x)^4/(d*cos(a+b*x))^(7/2),x,4,2/5*sin(a+b*x)^3/(b*d*(d*cos(a+b*x))^(5/2))-12/5*sin(a+b*x)/(b*d^3*sqrt(d*cos(a+b*x)))+24/5*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*d^4*sqrt(cos(a+b*x)))],
[sin(a+b*x)^4/(d*cos(a+b*x))^(9/2),x,4,-4/7*sin(a+b*x)/(b*d^3*(d*cos(a+b*x))^(3/2))+2/7*sin(a+b*x)^3/(b*d*(d*cos(a+b*x))^(7/2))+8/7*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*d^4*sqrt(d*cos(a+b*x)))],
[cos(a+b*x)^(3/2)*sin(a+b*x)^5,x,3,-2/5*cos(a+b*x)^(5/2)/b+4/9*cos(a+b*x)^(9/2)/b-2/13*cos(a+b*x)^(13/2)/b],

# m<0
[(d*cos(a+b*x))^(9/2)*csc(a+b*x),x,7,d^(9/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b-d^(9/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b+2/3*d^3*(d*cos(a+b*x))^(3/2)/b+2/7*d*(d*cos(a+b*x))^(7/2)/b],
[(d*cos(a+b*x))^(7/2)*csc(a+b*x),x,7,-d^(7/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b-d^(7/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b+2/5*d*(d*cos(a+b*x))^(5/2)/b+2*d^3*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(5/2)*csc(a+b*x),x,6,d^(5/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b-d^(5/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b+2/3*d*(d*cos(a+b*x))^(3/2)/b],
[(d*cos(a+b*x))^(3/2)*csc(a+b*x),x,6,-d^(3/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b-d^(3/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b+2*d*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(1/2)*csc(a+b*x),x,5,arctan(sqrt(d*cos(a+b*x))/sqrt(d))*sqrt(d)/b-arctanh(sqrt(d*cos(a+b*x))/sqrt(d))*sqrt(d)/b],
[csc(a+b*x)/(d*cos(a+b*x))^(1/2),x,5,-arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*sqrt(d))-arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*sqrt(d))],
[csc(a+b*x)/(d*cos(a+b*x))^(3/2),x,6,arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(3/2))-arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(3/2))+2/(b*d*sqrt(d*cos(a+b*x)))],
[csc(a+b*x)/(d*cos(a+b*x))^(5/2),x,6,-arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(5/2))-arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(5/2))+2/3/(b*d*(d*cos(a+b*x))^(3/2))],
[csc(a+b*x)/(d*cos(a+b*x))^(7/2),x,7,arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(7/2))-arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(7/2))+2/5/(b*d*(d*cos(a+b*x))^(5/2))+2/(b*d^3*sqrt(d*cos(a+b*x)))],
[csc(a+b*x)/(d*cos(a+b*x))^(9/2),x,7,-arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(9/2))-arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(9/2))+2/7/(b*d*(d*cos(a+b*x))^(7/2))+2/3/(b*d^3*(d*cos(a+b*x))^(3/2))],
[(d*cos(a+b*x))^(11/2)*csc(a+b*x)^2,x,5,-d*(d*cos(a+b*x))^(9/2)*csc(a+b*x)/b-9/7*d^3*(d*cos(a+b*x))^(5/2)*sin(a+b*x)/b-15/7*d^6*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))-15/7*d^5*sin(a+b*x)*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(9/2)*csc(a+b*x)^2,x,4,-d*(d*cos(a+b*x))^(7/2)*csc(a+b*x)/b-7/5*d^3*(d*cos(a+b*x))^(3/2)*sin(a+b*x)/b-21/5*d^4*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[(d*cos(a+b*x))^(7/2)*csc(a+b*x)^2,x,4,-d*(d*cos(a+b*x))^(5/2)*csc(a+b*x)/b-5/3*d^4*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))-5/3*d^3*sin(a+b*x)*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(5/2)*csc(a+b*x)^2,x,3,-d*(d*cos(a+b*x))^(3/2)*csc(a+b*x)/b-3*d^2*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[(d*cos(a+b*x))^(3/2)*csc(a+b*x)^2,x,3,-d^2*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))-d*csc(a+b*x)*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(1/2)*csc(a+b*x)^2,x,3,-(d*cos(a+b*x))^(3/2)*csc(a+b*x)/(b*d)-sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*sqrt(cos(a+b*x)))],
[csc(a+b*x)^2/(d*cos(a+b*x))^(1/2),x,3,sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*sqrt(d*cos(a+b*x)))-csc(a+b*x)*sqrt(d*cos(a+b*x))/(b*d)],
[csc(a+b*x)^2/(d*cos(a+b*x))^(3/2),x,4,-csc(a+b*x)/(b*d*sqrt(d*cos(a+b*x)))+3*sin(a+b*x)/(b*d*sqrt(d*cos(a+b*x)))-3*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*d^2*sqrt(cos(a+b*x)))],
[csc(a+b*x)^2/(d*cos(a+b*x))^(5/2),x,4,-csc(a+b*x)/(b*d*(d*cos(a+b*x))^(3/2))+5/3*sin(a+b*x)/(b*d*(d*cos(a+b*x))^(3/2))+5/3*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticF(sin(1/2*(a+b*x)),sqrt(2))*sqrt(cos(a+b*x))/(b*d^2*sqrt(d*cos(a+b*x)))],
[csc(a+b*x)^2/(d*cos(a+b*x))^(7/2),x,5,-csc(a+b*x)/(b*d*(d*cos(a+b*x))^(5/2))+7/5*sin(a+b*x)/(b*d*(d*cos(a+b*x))^(5/2))+21/5*sin(a+b*x)/(b*d^3*sqrt(d*cos(a+b*x)))-21/5*sqrt(cos(1/2*(a+b*x))^2)/cos(1/2*(a+b*x))*EllipticE(sin(1/2*(a+b*x)),sqrt(2))*sqrt(d*cos(a+b*x))/(b*d^4*sqrt(cos(a+b*x)))],
[(d*cos(a+b*x))^(11/2)*csc(a+b*x)^3,x,8,9/4*d^(11/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b+9/4*d^(11/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b-9/10*d^3*(d*cos(a+b*x))^(5/2)/b-1/2*d*(d*cos(a+b*x))^(9/2)*csc(a+b*x)^2/b-9/2*d^5*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(9/2)*csc(a+b*x)^3,x,7,-7/4*d^(9/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b+7/4*d^(9/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b-7/6*d^3*(d*cos(a+b*x))^(3/2)/b-1/2*d*(d*cos(a+b*x))^(7/2)*csc(a+b*x)^2/b],
[(d*cos(a+b*x))^(7/2)*csc(a+b*x)^3,x,7,5/4*d^(7/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b+5/4*d^(7/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b-1/2*d*(d*cos(a+b*x))^(5/2)*csc(a+b*x)^2/b-5/2*d^3*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(5/2)*csc(a+b*x)^3,x,6,-3/4*d^(5/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b+3/4*d^(5/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b-1/2*d*(d*cos(a+b*x))^(3/2)*csc(a+b*x)^2/b],
[(d*cos(a+b*x))^(3/2)*csc(a+b*x)^3,x,6,1/4*d^(3/2)*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/b+1/4*d^(3/2)*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/b-1/2*d*csc(a+b*x)^2*sqrt(d*cos(a+b*x))/b],
[(d*cos(a+b*x))^(1/2)*csc(a+b*x)^3,x,6,-1/2*(d*cos(a+b*x))^(3/2)*csc(a+b*x)^2/(b*d)+1/4*arctan(sqrt(d*cos(a+b*x))/sqrt(d))*sqrt(d)/b-1/4*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))*sqrt(d)/b],
[csc(a+b*x)^3/(d*cos(a+b*x))^(1/2),x,6,-3/4*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*sqrt(d))-3/4*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*sqrt(d))-1/2*csc(a+b*x)^2*sqrt(d*cos(a+b*x))/(b*d)],
[csc(a+b*x)^3/(d*cos(a+b*x))^(3/2),x,7,5/4*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(3/2))-5/4*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(3/2))+5/2/(b*d*sqrt(d*cos(a+b*x)))-1/2*csc(a+b*x)^2/(b*d*sqrt(d*cos(a+b*x)))],
[csc(a+b*x)^3/(d*cos(a+b*x))^(5/2),x,7,-7/4*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(5/2))-7/4*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(5/2))+7/6/(b*d*(d*cos(a+b*x))^(3/2))-1/2*csc(a+b*x)^2/(b*d*(d*cos(a+b*x))^(3/2))],
[csc(a+b*x)^3/(d*cos(a+b*x))^(7/2),x,8,9/4*arctan(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(7/2))-9/4*arctanh(sqrt(d*cos(a+b*x))/sqrt(d))/(b*d^(7/2))+9/10/(b*d*(d*cos(a+b*x))^(5/2))-1/2*csc(a+b*x)^2/(b*d*(d*cos(a+b*x))^(5/2))+9/2/(b*d^3*sqrt(d*cos(a+b*x)))],

# Integrands of the form Sin[e+f x]^m (b Cos[e+f x])^(n/5)
[(d*cos(a+b*x))^(1/5)*sin(a+b*x),x,2,-5/6*(d*cos(a+b*x))^(6/5)/(b*d)],

# Integrands of the form (a Sin[e+f x])^(m/2) Cos[e+f x]^n

# m>0
[cos(x)^3*sqrt(sin(x)),x,3,2/3*sin(x)^(3/2)-2/7*sin(x)^(7/2)],
[cos(x)^3*sin(x)^(3/2),x,3,2/5*sin(x)^(5/2)-2/9*sin(x)^(9/2)],
[cos(x)^3*sin(x)^(5/2),x,3,2/7*sin(x)^(7/2)-2/11*sin(x)^(11/2)],

# m<0
[cos(x)^3/sqrt(sin(x)),x,3,-2/5*sin(x)^(5/2)+2*sqrt(sin(x))],

# Integrands of the form (a Sin[e+f x])^(m/2) (b Cos[e+f x])^(n/2)

# m>0
[(d*cos(a+b*x))^(9/2)*(c*sin(a+b*x))^(1/2),x,4,7/30*d^3*(d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^(3/2)/(b*c)+1/5*d*(d*cos(a+b*x))^(7/2)*(c*sin(a+b*x))^(3/2)/(b*c)+7/20*d^4*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*sqrt(sin(2*a+2*b*x)))],
[(d*cos(a+b*x))^(5/2)*(c*sin(a+b*x))^(1/2),x,3,1/3*d*(d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^(3/2)/(b*c)+1/2*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*sqrt(sin(2*a+2*b*x)))],
[(d*cos(a+b*x))^(1/2)*(c*sin(a+b*x))^(1/2),x,2,sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*sqrt(sin(2*a+2*b*x)))],
[(c*sin(a+b*x))^(1/2)/(d*cos(a+b*x))^(3/2),x,3,2*(c*sin(a+b*x))^(3/2)/(b*c*d*sqrt(d*cos(a+b*x)))-2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*d^2*sqrt(sin(2*a+2*b*x)))],
[(c*sin(a+b*x))^(1/2)/(d*cos(a+b*x))^(7/2),x,4,2/5*(c*sin(a+b*x))^(3/2)/(b*c*d*(d*cos(a+b*x))^(5/2))+4/5*(c*sin(a+b*x))^(3/2)/(b*c*d^3*sqrt(d*cos(a+b*x)))-4/5*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*d^4*sqrt(sin(2*a+2*b*x)))],
[(d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^(1/2),x,11,-1/4*d^(3/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/(sqrt(c)*sqrt(d*cos(a+b*x))))*sqrt(c)/(b*sqrt(2))+1/4*d^(3/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/(sqrt(c)*sqrt(d*cos(a+b*x))))*sqrt(c)/(b*sqrt(2))+1/8*d^(3/2)*log(sqrt(c)-sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/sqrt(d*cos(a+b*x))+sqrt(c)*tan(a+b*x))*sqrt(c)/(b*sqrt(2))-1/8*d^(3/2)*log(sqrt(c)+sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/sqrt(d*cos(a+b*x))+sqrt(c)*tan(a+b*x))*sqrt(c)/(b*sqrt(2))+1/2*d*(c*sin(a+b*x))^(3/2)*sqrt(d*cos(a+b*x))/(b*c)],
[(c*sin(a+b*x))^(1/2)/(d*cos(a+b*x))^(1/2),x,10,-arctan(1-sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/(sqrt(c)*sqrt(d*cos(a+b*x))))*sqrt(c)/(b*sqrt(2)*sqrt(d))+arctan(1+sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/(sqrt(c)*sqrt(d*cos(a+b*x))))*sqrt(c)/(b*sqrt(2)*sqrt(d))+1/2*log(sqrt(c)-sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/sqrt(d*cos(a+b*x))+sqrt(c)*tan(a+b*x))*sqrt(c)/(b*sqrt(2)*sqrt(d))-1/2*log(sqrt(c)+sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/sqrt(d*cos(a+b*x))+sqrt(c)*tan(a+b*x))*sqrt(c)/(b*sqrt(2)*sqrt(d))],
[(c*sin(a+b*x))^(1/2)/(d*cos(a+b*x))^(5/2),x,1,2/3*(c*sin(a+b*x))^(3/2)/(b*c*d*(d*cos(a+b*x))^(3/2))],
[(c*sin(a+b*x))^(1/2)/(d*cos(a+b*x))^(9/2),x,2,2/7*(c*sin(a+b*x))^(3/2)/(b*c*d*(d*cos(a+b*x))^(7/2))+8/21*(c*sin(a+b*x))^(3/2)/(b*c*d^3*(d*cos(a+b*x))^(3/2))],
[(c*sin(a+b*x))^(1/2)/(d*cos(a+b*x))^(13/2),x,3,2/11*(c*sin(a+b*x))^(3/2)/(b*c*d*(d*cos(a+b*x))^(11/2))+16/77*(c*sin(a+b*x))^(3/2)/(b*c*d^3*(d*cos(a+b*x))^(7/2))+64/231*(c*sin(a+b*x))^(3/2)/(b*c*d^5*(d*cos(a+b*x))^(3/2))],
[(d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^(3/2),x,4,-1/3*c*(d*cos(a+b*x))^(5/2)*sqrt(c*sin(a+b*x))/(b*d)+1/6*c*d*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/b+1/12*c^2*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[(c*sin(a+b*x))^(3/2)/(d*cos(a+b*x))^(1/2),x,3,-c*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*d)+1/2*c^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[(c*sin(a+b*x))^(3/2)/(d*cos(a+b*x))^(5/2),x,3,2/3*c*sqrt(c*sin(a+b*x))/(b*d*(d*cos(a+b*x))^(3/2))-1/3*c^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*d^2*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[(c*sin(a+b*x))^(3/2)/(d*cos(a+b*x))^(9/2),x,4,2/7*c*sqrt(c*sin(a+b*x))/(b*d*(d*cos(a+b*x))^(7/2))-2/21*c*sqrt(c*sin(a+b*x))/(b*d^3*(d*cos(a+b*x))^(3/2))-2/21*c^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*d^4*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[(d*cos(a+b*x))^(1/2)*(c*sin(a+b*x))^(3/2),x,11,1/4*c^(3/2)*arctan(1-sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/(sqrt(d)*sqrt(c*sin(a+b*x))))*sqrt(d)/(b*sqrt(2))-1/4*c^(3/2)*arctan(1+sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/(sqrt(d)*sqrt(c*sin(a+b*x))))*sqrt(d)/(b*sqrt(2))-1/8*c^(3/2)*log(sqrt(d)+cot(a+b*x)*sqrt(d)-sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/sqrt(c*sin(a+b*x)))*sqrt(d)/(b*sqrt(2))+1/8*c^(3/2)*log(sqrt(d)+cot(a+b*x)*sqrt(d)+sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/sqrt(c*sin(a+b*x)))*sqrt(d)/(b*sqrt(2))-1/2*c*(d*cos(a+b*x))^(3/2)*sqrt(c*sin(a+b*x))/(b*d)],
[(c*sin(a+b*x))^(3/2)/(d*cos(a+b*x))^(3/2),x,11,-c^(3/2)*arctan(1-sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/(sqrt(d)*sqrt(c*sin(a+b*x))))/(b*d^(3/2)*sqrt(2))+c^(3/2)*arctan(1+sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/(sqrt(d)*sqrt(c*sin(a+b*x))))/(b*d^(3/2)*sqrt(2))+1/2*c^(3/2)*log(sqrt(d)+cot(a+b*x)*sqrt(d)-sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/sqrt(c*sin(a+b*x)))/(b*d^(3/2)*sqrt(2))-1/2*c^(3/2)*log(sqrt(d)+cot(a+b*x)*sqrt(d)+sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/sqrt(c*sin(a+b*x)))/(b*d^(3/2)*sqrt(2))+2*c*sqrt(c*sin(a+b*x))/(b*d*sqrt(d*cos(a+b*x)))],
[(c*sin(a+b*x))^(3/2)/(d*cos(a+b*x))^(7/2),x,1,2/5*(c*sin(a+b*x))^(5/2)/(b*c*d*(d*cos(a+b*x))^(5/2))],
[(c*sin(a+b*x))^(3/2)/(d*cos(a+b*x))^(11/2),x,3,2/9*c*sqrt(c*sin(a+b*x))/(b*d*(d*cos(a+b*x))^(9/2))-2/45*c*sqrt(c*sin(a+b*x))/(b*d^3*(d*cos(a+b*x))^(5/2))-8/45*c*sqrt(c*sin(a+b*x))/(b*d^5*sqrt(d*cos(a+b*x)))],
[(c*sin(a+b*x))^(3/2)/(d*cos(a+b*x))^(15/2),x,4,2/13*c*sqrt(c*sin(a+b*x))/(b*d*(d*cos(a+b*x))^(13/2))-2/117*c*sqrt(c*sin(a+b*x))/(b*d^3*(d*cos(a+b*x))^(9/2))-16/585*c*sqrt(c*sin(a+b*x))/(b*d^5*(d*cos(a+b*x))^(5/2))-64/585*c*sqrt(c*sin(a+b*x))/(b*d^7*sqrt(d*cos(a+b*x)))],
[(d*cos(a+b*x))^(9/2)*(c*sin(a+b*x))^(5/2),x,5,1/20*c*d^3*(d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^(3/2)/b+3/70*c*d*(d*cos(a+b*x))^(7/2)*(c*sin(a+b*x))^(3/2)/b-1/7*c*(d*cos(a+b*x))^(11/2)*(c*sin(a+b*x))^(3/2)/(b*d)+3/40*c^2*d^4*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*sqrt(sin(2*a+2*b*x)))],
[(d*cos(a+b*x))^(5/2)*(c*sin(a+b*x))^(5/2),x,4,1/10*c*d*(d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^(3/2)/b-1/5*c*(d*cos(a+b*x))^(7/2)*(c*sin(a+b*x))^(3/2)/(b*d)+3/20*c^2*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*sqrt(sin(2*a+2*b*x)))],
[(d*cos(a+b*x))^(1/2)*(c*sin(a+b*x))^(5/2),x,3,-1/3*c*(d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^(3/2)/(b*d)+1/2*c^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*sqrt(sin(2*a+2*b*x)))],
[(c*sin(a+b*x))^(5/2)/(d*cos(a+b*x))^(3/2),x,3,2*c*(c*sin(a+b*x))^(3/2)/(b*d*sqrt(d*cos(a+b*x)))-3*c^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*d^2*sqrt(sin(2*a+2*b*x)))],
[(c*sin(a+b*x))^(5/2)/(d*cos(a+b*x))^(7/2),x,4,2/5*c*(c*sin(a+b*x))^(3/2)/(b*d*(d*cos(a+b*x))^(5/2))-6/5*c*(c*sin(a+b*x))^(3/2)/(b*d^3*sqrt(d*cos(a+b*x)))+6/5*c^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*d^4*sqrt(sin(2*a+2*b*x)))],
[(c*sin(a+b*x))^(5/2)/(d*cos(a+b*x))^(11/2),x,5,2/9*c*(c*sin(a+b*x))^(3/2)/(b*d*(d*cos(a+b*x))^(9/2))-2/15*c*(c*sin(a+b*x))^(3/2)/(b*d^3*(d*cos(a+b*x))^(5/2))-4/15*c*(c*sin(a+b*x))^(3/2)/(b*d^5*sqrt(d*cos(a+b*x)))+4/15*c^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticE(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*d^6*sqrt(sin(2*a+2*b*x)))],
[(c*sin(a+b*x))^(5/2)/(d*cos(a+b*x))^(1/2),x,11,-3/4*c^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/(sqrt(c)*sqrt(d*cos(a+b*x))))/(b*sqrt(2)*sqrt(d))+3/4*c^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/(sqrt(c)*sqrt(d*cos(a+b*x))))/(b*sqrt(2)*sqrt(d))+3/8*c^(5/2)*log(sqrt(c)-sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/sqrt(d*cos(a+b*x))+sqrt(c)*tan(a+b*x))/(b*sqrt(2)*sqrt(d))-3/8*c^(5/2)*log(sqrt(c)+sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/sqrt(d*cos(a+b*x))+sqrt(c)*tan(a+b*x))/(b*sqrt(2)*sqrt(d))-1/2*c*(c*sin(a+b*x))^(3/2)*sqrt(d*cos(a+b*x))/(b*d)],
[(c*sin(a+b*x))^(5/2)/(d*cos(a+b*x))^(5/2),x,11,2/3*c*(c*sin(a+b*x))^(3/2)/(b*d*(d*cos(a+b*x))^(3/2))+c^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/(sqrt(c)*sqrt(d*cos(a+b*x))))/(b*d^(5/2)*sqrt(2))-c^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/(sqrt(c)*sqrt(d*cos(a+b*x))))/(b*d^(5/2)*sqrt(2))-1/2*c^(5/2)*log(sqrt(c)-sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/sqrt(d*cos(a+b*x))+sqrt(c)*tan(a+b*x))/(b*d^(5/2)*sqrt(2))+1/2*c^(5/2)*log(sqrt(c)+sqrt(2)*sqrt(d)*sqrt(c*sin(a+b*x))/sqrt(d*cos(a+b*x))+sqrt(c)*tan(a+b*x))/(b*d^(5/2)*sqrt(2))],
[(c*sin(a+b*x))^(5/2)/(d*cos(a+b*x))^(9/2),x,1,2/7*(c*sin(a+b*x))^(7/2)/(b*c*d*(d*cos(a+b*x))^(7/2))],
[(c*sin(a+b*x))^(5/2)/(d*cos(a+b*x))^(13/2),x,3,2/11*c*(c*sin(a+b*x))^(3/2)/(b*d*(d*cos(a+b*x))^(11/2))-6/77*c*(c*sin(a+b*x))^(3/2)/(b*d^3*(d*cos(a+b*x))^(7/2))-8/77*c*(c*sin(a+b*x))^(3/2)/(b*d^5*(d*cos(a+b*x))^(3/2))],
[(c*sin(a+b*x))^(5/2)/(d*cos(a+b*x))^(17/2),x,4,2/15*c*(c*sin(a+b*x))^(3/2)/(b*d*(d*cos(a+b*x))^(15/2))-2/55*c*(c*sin(a+b*x))^(3/2)/(b*d^3*(d*cos(a+b*x))^(11/2))-16/385*c*(c*sin(a+b*x))^(3/2)/(b*d^5*(d*cos(a+b*x))^(7/2))-64/1155*c*(c*sin(a+b*x))^(3/2)/(b*d^7*(d*cos(a+b*x))^(3/2))],
[sin(a+b*x)^(7/2)/cos(a+b*x)^(7/2),x,12,2/5*sin(a+b*x)^(5/2)/(b*cos(a+b*x)^(5/2))+arctan(1-sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))-arctan(1+sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))-1/2*log(1+cot(a+b*x)-sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))+1/2*log(1+cot(a+b*x)+sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))-2*sqrt(sin(a+b*x))/(b*sqrt(cos(a+b*x)))],
[sin(x)^(3/2)/cos(x)^(7/2),x,1,2/5*sin(x)^(5/2)/cos(x)^(5/2)],
[sqrt(sin(x))/sqrt(cos(x)),x,10,-arctan(1-sqrt(2)*sqrt(sin(x))/sqrt(cos(x)))/sqrt(2)+arctan(1+sqrt(2)*sqrt(sin(x))/sqrt(cos(x)))/sqrt(2)+1/2*log(1-sqrt(2)*sqrt(sin(x))/sqrt(cos(x))+tan(x))/sqrt(2)-1/2*log(1+sqrt(2)*sqrt(sin(x))/sqrt(cos(x))+tan(x))/sqrt(2)],
[sin(x)^(5/2)/sqrt(cos(x)),x,11,-3/4*arctan(1-sqrt(2)*sqrt(sin(x))/sqrt(cos(x)))/sqrt(2)+3/4*arctan(1+sqrt(2)*sqrt(sin(x))/sqrt(cos(x)))/sqrt(2)+3/8*log(1-sqrt(2)*sqrt(sin(x))/sqrt(cos(x))+tan(x))/sqrt(2)-3/8*log(1+sqrt(2)*sqrt(sin(x))/sqrt(cos(x))+tan(x))/sqrt(2)-1/2*sin(x)^(3/2)*sqrt(cos(x))],

# m<0
[(d*cos(a+b*x))^(7/2)/(c*sin(a+b*x))^(1/2),x,4,1/3*d*(d*cos(a+b*x))^(5/2)*sqrt(c*sin(a+b*x))/(b*c)+5/6*d^3*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*c)+5/12*d^4*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[(d*cos(a+b*x))^(3/2)/(c*sin(a+b*x))^(1/2),x,3,d*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x))/(b*c)+1/2*d^2*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[1/((d*cos(a+b*x))^(1/2)*(c*sin(a+b*x))^(1/2)),x,2,sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[1/((d*cos(a+b*x))^(5/2)*(c*sin(a+b*x))^(1/2)),x,3,2/3*sqrt(c*sin(a+b*x))/(b*c*d*(d*cos(a+b*x))^(3/2))+2/3*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*d^2*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[1/((d*cos(a+b*x))^(9/2)*(c*sin(a+b*x))^(1/2)),x,4,2/7*sqrt(c*sin(a+b*x))/(b*c*d*(d*cos(a+b*x))^(7/2))+4/7*sqrt(c*sin(a+b*x))/(b*c*d^3*(d*cos(a+b*x))^(3/2))+4/7*sqrt(cos(a-1/4*Pi+b*x)^2)/cos(a-1/4*Pi+b*x)*EllipticF(sin(a-1/4*Pi+b*x),sqrt(2))*sqrt(sin(2*a+2*b*x))/(b*d^4*sqrt(d*cos(a+b*x))*sqrt(c*sin(a+b*x)))],
[(d*cos(a+b*x))^(1/2)/(c*sin(a+b*x))^(1/2),x,10,arctan(1-sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/(sqrt(d)*sqrt(c*sin(a+b*x))))*sqrt(d)/(b*sqrt(2)*sqrt(c))-arctan(1+sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/(sqrt(d)*sqrt(c*sin(a+b*x))))*sqrt(d)/(b*sqrt(2)*sqrt(c))-1/2*log(sqrt(d)+cot(a+b*x)*sqrt(d)-sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/sqrt(c*sin(a+b*x)))*sqrt(d)/(b*sqrt(2)*sqrt(c))+1/2*log(sqrt(d)+cot(a+b*x)*sqrt(d)+sqrt(2)*sqrt(c)*sqrt(d*cos(a+b*x))/sqrt(c*sin(a+b*x)))*sqrt(d)/(b*sqrt(2)*sqrt(c))],
[1/((d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^(1/2)),x,1,2*sqrt(c*sin(a+b*x))/(b*c*d*sqrt(d*cos(a+b*x)))],
[1/((d*cos(a+b*x))^(7/2)*(c*sin(a+b*x))^(1/2)),x,2,2/5*sqrt(c*sin(a+b*x))/(b*c*d*(d*cos(a+b*x))^(5/2))+8/5*sqrt(c*sin(a+b*x))/(b*c*d^3*sqrt(d*cos(a+b*x)))],
[1/((d*cos(a+b*x))^(11/2)*(c*sin(a+b*x))^(1/2)),x,3,2/9*sqrt(c*sin(a+b*x))/(b*c*d*(d*cos(a+b*x))^(9/2))+16/45*sqrt(c*sin(a+b*x))/(b*c*d^3*(d*cos(a+b*x))^(5/2))+64/45*sqrt(c*sin(a+b*x))/(b*c*d^5*sqrt(d*cos(a+b*x)))],
[cos(a+b*x)^(1/2)/sin(a+b*x)^(1/2),x,10,arctan(1-sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))-arctan(1+sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))-1/2*log(1+cot(a+b*x)-sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))+1/2*log(1+cot(a+b*x)+sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))],
[cos(a+b*x)^(3/2)/sin(a+b*x)^(3/2),x,11,arctan(1-sqrt(2)*sqrt(sin(a+b*x))/sqrt(cos(a+b*x)))/(b*sqrt(2))-arctan(1+sqrt(2)*sqrt(sin(a+b*x))/sqrt(cos(a+b*x)))/(b*sqrt(2))-1/2*log(1-sqrt(2)*sqrt(sin(a+b*x))/sqrt(cos(a+b*x))+tan(a+b*x))/(b*sqrt(2))+1/2*log(1+sqrt(2)*sqrt(sin(a+b*x))/sqrt(cos(a+b*x))+tan(a+b*x))/(b*sqrt(2))-2*sqrt(cos(a+b*x))/(b*sqrt(sin(a+b*x)))],
[cos(a+b*x)^(5/2)/sin(a+b*x)^(5/2),x,11,-2/3*cos(a+b*x)^(3/2)/(b*sin(a+b*x)^(3/2))-arctan(1-sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))+arctan(1+sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))+1/2*log(1+cot(a+b*x)-sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))-1/2*log(1+cot(a+b*x)+sqrt(2)*sqrt(cos(a+b*x))/sqrt(sin(a+b*x)))/(b*sqrt(2))],
[cos(a+b*x)^(7/2)/sin(a+b*x)^(7/2),x,12,-2/5*cos(a+b*x)^(5/2)/(b*sin(a+b*x)^(5/2))-arctan(1-sqrt(2)*sqrt(sin(a+b*x))/sqrt(cos(a+b*x)))/(b*sqrt(2))+arctan(1+sqrt(2)*sqrt(sin(a+b*x))/sqrt(cos(a+b*x)))/(b*sqrt(2))+1/2*log(1-sqrt(2)*sqrt(sin(a+b*x))/sqrt(cos(a+b*x))+tan(a+b*x))/(b*sqrt(2))-1/2*log(1+sqrt(2)*sqrt(sin(a+b*x))/sqrt(cos(a+b*x))+tan(a+b*x))/(b*sqrt(2))+2*sqrt(cos(a+b*x))/(b*sqrt(sin(a+b*x)))],

# Integrands of the form (a Sin[e+f x])^(m/3) Cos[e+f x])^n

# m>0
[cos(e+f*x)^4*(b*sin(e+f*x))^(1/3),x,1,3/4*cos(e+f*x)*hypergeom([-3/2,2/3],[5/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(4/3)/(b*f*sqrt(cos(e+f*x)^2))],
[cos(e+f*x)^2*(b*sin(e+f*x))^(1/3),x,1,3/4*cos(e+f*x)*hypergeom([-1/2,2/3],[5/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(4/3)/(b*f*sqrt(cos(e+f*x)^2))],
[(b*sin(e+f*x))^(1/3),x,1,3/4*cos(e+f*x)*hypergeom([1/2,2/3],[5/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(4/3)/(b*f*sqrt(cos(e+f*x)^2))],
[sec(e+f*x)^2*(b*sin(e+f*x))^(1/3),x,1,3/4*hypergeom([2/3,3/2],[5/3],sin(e+f*x)^2)*sec(e+f*x)*(b*sin(e+f*x))^(4/3)*sqrt(cos(e+f*x)^2)/(b*f)],
[sec(e+f*x)^4*(b*sin(e+f*x))^(1/3),x,1,3/4*hypergeom([2/3,5/2],[5/3],sin(e+f*x)^2)*sec(e+f*x)*(b*sin(e+f*x))^(4/3)*sqrt(cos(e+f*x)^2)/(b*f)],
[cos(e+f*x)^4*(b*sin(e+f*x))^(5/3),x,1,3/8*cos(e+f*x)*hypergeom([-3/2,4/3],[7/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(8/3)/(b*f*sqrt(cos(e+f*x)^2))],
[cos(e+f*x)^2*(b*sin(e+f*x))^(5/3),x,1,3/8*cos(e+f*x)*hypergeom([-1/2,4/3],[7/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(8/3)/(b*f*sqrt(cos(e+f*x)^2))],
[(b*sin(e+f*x))^(5/3),x,1,3/8*cos(e+f*x)*hypergeom([1/2,4/3],[7/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(8/3)/(b*f*sqrt(cos(e+f*x)^2))],
[sec(e+f*x)^2*(b*sin(e+f*x))^(5/3),x,1,3/8*hypergeom([4/3,3/2],[7/3],sin(e+f*x)^2)*sec(e+f*x)*(b*sin(e+f*x))^(8/3)*sqrt(cos(e+f*x)^2)/(b*f)],
[sec(e+f*x)^4*(b*sin(e+f*x))^(5/3),x,1,3/8*hypergeom([4/3,5/2],[7/3],sin(e+f*x)^2)*sec(e+f*x)*(b*sin(e+f*x))^(8/3)*sqrt(cos(e+f*x)^2)/(b*f)],

# m<0
[cos(e+f*x)^4/(b*sin(e+f*x))^(1/3),x,1,3/2*cos(e+f*x)*hypergeom([-3/2,1/3],[4/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(2/3)/(b*f*sqrt(cos(e+f*x)^2))],
[cos(e+f*x)^2/(b*sin(e+f*x))^(1/3),x,1,3/2*cos(e+f*x)*hypergeom([-1/2,1/3],[4/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(2/3)/(b*f*sqrt(cos(e+f*x)^2))],
[1/(b*sin(e+f*x))^(1/3),x,1,3/2*cos(e+f*x)*hypergeom([1/3,1/2],[4/3],sin(e+f*x)^2)*(b*sin(e+f*x))^(2/3)/(b*f*sqrt(cos(e+f*x)^2))],
[sec(e+f*x)^2/(b*sin(e+f*x))^(1/3),x,1,3/2*hypergeom([1/3,3/2],[4/3],sin(e+f*x)^2)*sec(e+f*x)*(b*sin(e+f*x))^(2/3)*sqrt(cos(e+f*x)^2)/(b*f)],
[sec(e+f*x)^4/(b*sin(e+f*x))^(1/3),x,1,3/2*hypergeom([1/3,5/2],[4/3],sin(e+f*x)^2)*sec(e+f*x)*(b*sin(e+f*x))^(2/3)*sqrt(cos(e+f*x)^2)/(b*f)],
[cos(e+f*x)^4/(b*sin(e+f*x))^(5/3),x,1,-3/2*cos(e+f*x)*hypergeom([-3/2,-1/3],[2/3],sin(e+f*x)^2)/(b*f*(b*sin(e+f*x))^(2/3)*sqrt(cos(e+f*x)^2))],
[cos(e+f*x)^2/(b*sin(e+f*x))^(5/3),x,1,-3/2*cos(e+f*x)*hypergeom([-1/2,-1/3],[2/3],sin(e+f*x)^2)/(b*f*(b*sin(e+f*x))^(2/3)*sqrt(cos(e+f*x)^2))],
[1/(b*sin(e+f*x))^(5/3),x,1,-3/2*cos(e+f*x)*hypergeom([-1/3,1/2],[2/3],sin(e+f*x)^2)/(b*f*(b*sin(e+f*x))^(2/3)*sqrt(cos(e+f*x)^2))],
[sec(e+f*x)^2/(b*sin(e+f*x))^(5/3),x,1,-3/2*hypergeom([-1/3,3/2],[2/3],sin(e+f*x)^2)*sec(e+f*x)*sqrt(cos(e+f*x)^2)/(b*f*(b*sin(e+f*x))^(2/3))],
[sec(e+f*x)^4/(b*sin(e+f*x))^(5/3),x,1,-3/2*hypergeom([-1/3,5/2],[2/3],sin(e+f*x)^2)*sec(e+f*x)*sqrt(cos(e+f*x)^2)/(b*f*(b*sin(e+f*x))^(2/3))],

# Integrands of the form (a Sin[e+f x])^(m/3) (b Cos[e+f x])^(n/3)

# m>0
[sin(a+b*x)^(1/3)/cos(a+b*x)^(1/3),x,8,-1/2*log(1+sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3))/b+1/4*log(1-sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3)+sin(a+b*x)^(4/3)/cos(a+b*x)^(4/3))/b-1/2*arctan((1-2*sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3))/sqrt(3))*sqrt(3)/b],
[sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3),x,11,arctan(sin(a+b*x)^(1/3)/cos(a+b*x)^(1/3))/b-1/2*arctan(-2*sin(a+b*x)^(1/3)/cos(a+b*x)^(1/3)+sqrt(3))/b+1/2*arctan(2*sin(a+b*x)^(1/3)/cos(a+b*x)^(1/3)+sqrt(3))/b+1/4*log(1+sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3)-sin(a+b*x)^(1/3)*sqrt(3)/cos(a+b*x)^(1/3))*sqrt(3)/b-1/4*log(1+sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3)+sin(a+b*x)^(1/3)*sqrt(3)/cos(a+b*x)^(1/3))*sqrt(3)/b],
[sin(a+b*x)^(4/3)/cos(a+b*x)^(4/3),x,12,arctan(cos(a+b*x)^(1/3)/sin(a+b*x)^(1/3))/b-1/2*arctan(-2*cos(a+b*x)^(1/3)/sin(a+b*x)^(1/3)+sqrt(3))/b+1/2*arctan(2*cos(a+b*x)^(1/3)/sin(a+b*x)^(1/3)+sqrt(3))/b+3*sin(a+b*x)^(1/3)/(b*cos(a+b*x)^(1/3))+1/4*log(1+cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3)-cos(a+b*x)^(1/3)*sqrt(3)/sin(a+b*x)^(1/3))*sqrt(3)/b-1/4*log(1+cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3)+cos(a+b*x)^(1/3)*sqrt(3)/sin(a+b*x)^(1/3))*sqrt(3)/b],
[sin(a+b*x)^(5/3)/cos(a+b*x)^(5/3),x,9,1/4*log(1+cos(a+b*x)^(4/3)/sin(a+b*x)^(4/3)-cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/b-1/2*log(1+cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/b+3/2*sin(a+b*x)^(2/3)/(b*cos(a+b*x)^(2/3))-1/2*arctan((1-2*cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/sqrt(3))*sqrt(3)/b],
[sin(a+b*x)^(7/3)/cos(a+b*x)^(7/3),x,9,1/2*log(1+sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3))/b-1/4*log(1-sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3)+sin(a+b*x)^(4/3)/cos(a+b*x)^(4/3))/b+3/4*sin(a+b*x)^(4/3)/(b*cos(a+b*x)^(4/3))+1/2*arctan((1-2*sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3))/sqrt(3))*sqrt(3)/b],

# m<0
[cos(a+b*x)^(1/3)/sin(a+b*x)^(1/3),x,8,-1/4*log(1+cos(a+b*x)^(4/3)/sin(a+b*x)^(4/3)-cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/b+1/2*log(1+cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/b+1/2*arctan((1-2*cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/sqrt(3))*sqrt(3)/b],
[cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3),x,11,-arctan(cos(a+b*x)^(1/3)/sin(a+b*x)^(1/3))/b+1/2*arctan(-2*cos(a+b*x)^(1/3)/sin(a+b*x)^(1/3)+sqrt(3))/b-1/2*arctan(2*cos(a+b*x)^(1/3)/sin(a+b*x)^(1/3)+sqrt(3))/b-1/4*log(1+cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3)-cos(a+b*x)^(1/3)*sqrt(3)/sin(a+b*x)^(1/3))*sqrt(3)/b+1/4*log(1+cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3)+cos(a+b*x)^(1/3)*sqrt(3)/sin(a+b*x)^(1/3))*sqrt(3)/b],
[cos(a+b*x)^(4/3)/sin(a+b*x)^(4/3),x,12,-arctan(sin(a+b*x)^(1/3)/cos(a+b*x)^(1/3))/b+1/2*arctan(-2*sin(a+b*x)^(1/3)/cos(a+b*x)^(1/3)+sqrt(3))/b-1/2*arctan(2*sin(a+b*x)^(1/3)/cos(a+b*x)^(1/3)+sqrt(3))/b-3*cos(a+b*x)^(1/3)/(b*sin(a+b*x)^(1/3))-1/4*log(1+sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3)-sin(a+b*x)^(1/3)*sqrt(3)/cos(a+b*x)^(1/3))*sqrt(3)/b+1/4*log(1+sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3)+sin(a+b*x)^(1/3)*sqrt(3)/cos(a+b*x)^(1/3))*sqrt(3)/b],
[cos(a+b*x)^(5/3)/sin(a+b*x)^(5/3),x,9,1/2*log(1+sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3))/b-1/4*log(1-sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3)+sin(a+b*x)^(4/3)/cos(a+b*x)^(4/3))/b-3/2*cos(a+b*x)^(2/3)/(b*sin(a+b*x)^(2/3))+1/2*arctan((1-2*sin(a+b*x)^(2/3)/cos(a+b*x)^(2/3))/sqrt(3))*sqrt(3)/b],
[cos(a+b*x)^(7/3)/sin(a+b*x)^(7/3),x,9,1/4*log(1+cos(a+b*x)^(4/3)/sin(a+b*x)^(4/3)-cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/b-1/2*log(1+cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/b-3/4*cos(a+b*x)^(4/3)/(b*sin(a+b*x)^(4/3))-1/2*arctan((1-2*cos(a+b*x)^(2/3)/sin(a+b*x)^(2/3))/sqrt(3))*sqrt(3)/b],
[cos(x)^(2/3)/sin(x)^(8/3),x,1,-3/5*cos(x)^(5/3)/sin(x)^(5/3)],
[sin(x)^(2/3)/cos(x)^(8/3),x,1,3/5*sin(x)^(5/3)/cos(x)^(5/3)],

# Integrands of the form (a Sin[e+f x])^m (b Cos[e+f x])^n with m symbolic
[cos(e+f*x)^n*sin(e+f*x)^m,x,1,-cos(e+f*x)^(1+n)*hypergeom([1/2*(1-m),1/2*(1+n)],[1/2*(3+n)],cos(e+f*x)^2)*sin(e+f*x)^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/(f*(1+n))],
[(d*cos(e+f*x))^n*sin(e+f*x)^m,x,1,-(d*cos(e+f*x))^(1+n)*hypergeom([1/2*(1-m),1/2*(1+n)],[1/2*(3+n)],cos(e+f*x)^2)*sin(e+f*x)^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/(d*f*(1+n))],
[cos(e+f*x)^n*(b*sin(e+f*x))^m,x,1,-b*cos(e+f*x)^(1+n)*hypergeom([1/2*(1-m),1/2*(1+n)],[1/2*(3+n)],cos(e+f*x)^2)*(b*sin(e+f*x))^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/(f*(1+n))],
[(d*cos(e+f*x))^n*(b*sin(e+f*x))^m,x,1,-b*(d*cos(e+f*x))^(1+n)*hypergeom([1/2*(1-m),1/2*(1+n)],[1/2*(3+n)],cos(e+f*x)^2)*(b*sin(e+f*x))^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/(d*f*(1+n))],
[cos(a+b*x)^5*(c*sin(a+b*x))^m,x,3,(c*sin(a+b*x))^(1+m)/(b*c*(1+m))-2*(c*sin(a+b*x))^(3+m)/(b*c^3*(3+m))+(c*sin(a+b*x))^(5+m)/(b*c^5*(5+m))],
[cos(a+b*x)^3*(c*sin(a+b*x))^m,x,3,(c*sin(a+b*x))^(1+m)/(b*c*(1+m))-(c*sin(a+b*x))^(3+m)/(b*c^3*(3+m))],
[cos(a+b*x)*(c*sin(a+b*x))^m,x,2,(c*sin(a+b*x))^(1+m)/(b*c*(1+m))],
[sec(a+b*x)*(c*sin(a+b*x))^m,x,2,hypergeom([1,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*(1+m))],
[sec(a+b*x)^3*(c*sin(a+b*x))^m,x,2,hypergeom([2,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*(1+m))],
[cos(a+b*x)^4*(c*sin(a+b*x))^m,x,1,cos(a+b*x)*hypergeom([-3/2,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*(1+m)*sqrt(cos(a+b*x)^2))],
[cos(a+b*x)^2*(c*sin(a+b*x))^m,x,1,cos(a+b*x)*hypergeom([-1/2,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*(1+m)*sqrt(cos(a+b*x)^2))],
[(c*sin(a+b*x))^m,x,1,cos(a+b*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*(1+m)*sqrt(cos(a+b*x)^2))],
[sec(a+b*x)^2*(c*sin(a+b*x))^m,x,1,hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*sec(a+b*x)*(c*sin(a+b*x))^(1+m)*sqrt(cos(a+b*x)^2)/(b*c*(1+m))],
[sec(a+b*x)^4*(c*sin(a+b*x))^m,x,1,hypergeom([5/2,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*sec(a+b*x)*(c*sin(a+b*x))^(1+m)*sqrt(cos(a+b*x)^2)/(b*c*(1+m))],
[(d*cos(a+b*x))^(3/2)*(c*sin(a+b*x))^m,x,1,d*hypergeom([-1/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)*sqrt(d*cos(a+b*x))/(b*c*(1+m)*(cos(a+b*x)^2)^(1/4))],
[(d*cos(a+b*x))^(1/2)*(c*sin(a+b*x))^m,x,1,d*(cos(a+b*x)^2)^(1/4)*hypergeom([1/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*(1+m)*sqrt(d*cos(a+b*x)))],
[(c*sin(a+b*x))^m/(d*cos(a+b*x))^(1/2),x,1,d*(cos(a+b*x)^2)^(3/4)*hypergeom([3/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*(1+m)*(d*cos(a+b*x))^(3/2))],
[(c*sin(a+b*x))^m/(d*cos(a+b*x))^(3/2),x,1,(cos(a+b*x)^2)^(1/4)*hypergeom([5/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*d*(1+m)*sqrt(d*cos(a+b*x)))],
[(c*sin(a+b*x))^m/(d*cos(a+b*x))^(5/2),x,1,(cos(a+b*x)^2)^(3/4)*hypergeom([7/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*d*(1+m)*(d*cos(a+b*x))^(3/2))],

# Integrands of the form (a Sin[e+f x])^m (b Cos[e+f x])^n with n symbolic
[(d*cos(a+b*x))^n*sin(a+b*x)^5,x,3,-(d*cos(a+b*x))^(1+n)/(b*d*(1+n))+2*(d*cos(a+b*x))^(3+n)/(b*d^3*(3+n))-(d*cos(a+b*x))^(5+n)/(b*d^5*(5+n))],
[(d*cos(a+b*x))^n*sin(a+b*x)^3,x,3,-(d*cos(a+b*x))^(1+n)/(b*d*(1+n))+(d*cos(a+b*x))^(3+n)/(b*d^3*(3+n))],
[(d*cos(a+b*x))^n*sin(a+b*x),x,2,-(d*cos(a+b*x))^(1+n)/(b*d*(1+n))],
[(d*cos(a+b*x))^n*csc(a+b*x),x,2,-(d*cos(a+b*x))^(1+n)*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)/(b*d*(1+n))],
[(d*cos(a+b*x))^n*csc(a+b*x)^3,x,2,-(d*cos(a+b*x))^(1+n)*hypergeom([2,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)/(b*d*(1+n))],
[(d*cos(a+b*x))^n*csc(a+b*x)^5,x,2,-(d*cos(a+b*x))^(1+n)*hypergeom([3,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)/(b*d*(1+n))],
[(d*cos(a+b*x))^n*sin(a+b*x)^4,x,1,-(d*cos(a+b*x))^(1+n)*hypergeom([-3/2,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*sin(a+b*x)/(b*d*(1+n)*sqrt(sin(a+b*x)^2))],
[(d*cos(a+b*x))^n*sin(a+b*x)^2,x,1,-(d*cos(a+b*x))^(1+n)*hypergeom([-1/2,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*sin(a+b*x)/(b*d*(1+n)*sqrt(sin(a+b*x)^2))],
[(d*cos(a+b*x))^n,x,1,-(d*cos(a+b*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*sin(a+b*x)/(b*d*(1+n)*sqrt(sin(a+b*x)^2))],
[(d*cos(a+b*x))^n*csc(a+b*x)^2,x,1,-(d*cos(a+b*x))^(1+n)*csc(a+b*x)*hypergeom([3/2,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*sqrt(sin(a+b*x)^2)/(b*d*(1+n))],
[(d*cos(a+b*x))^n*csc(a+b*x)^4,x,1,-(d*cos(a+b*x))^(1+n)*csc(a+b*x)*hypergeom([5/2,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*sqrt(sin(a+b*x)^2)/(b*d*(1+n))],
[(d*cos(a+b*x))^n*(c*sin(a+b*x))^(5/2),x,1,-c*(d*cos(a+b*x))^(1+n)*hypergeom([-3/4,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*(c*sin(a+b*x))^(3/2)/(b*d*(1+n)*(sin(a+b*x)^2)^(3/4))],
[(d*cos(a+b*x))^n*(c*sin(a+b*x))^(3/2),x,1,-c*(d*cos(a+b*x))^(1+n)*hypergeom([-1/4,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*sqrt(c*sin(a+b*x))/(b*d*(1+n)*(sin(a+b*x)^2)^(1/4))],
[(d*cos(a+b*x))^n*(c*sin(a+b*x))^(1/2),x,1,-c*(d*cos(a+b*x))^(1+n)*hypergeom([1/4,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*(sin(a+b*x)^2)^(1/4)/(b*d*(1+n)*sqrt(c*sin(a+b*x)))],
[(d*cos(a+b*x))^n/(c*sin(a+b*x))^(1/2),x,1,-c*(d*cos(a+b*x))^(1+n)*hypergeom([3/4,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*(sin(a+b*x)^2)^(3/4)/(b*d*(1+n)*(c*sin(a+b*x))^(3/2))],
[(d*cos(a+b*x))^n/(c*sin(a+b*x))^(3/2),x,1,-(d*cos(a+b*x))^(1+n)*hypergeom([5/4,1/2*(1+n)],[1/2*(3+n)],cos(a+b*x)^2)*(sin(a+b*x)^2)^(1/4)/(b*c*d*(1+n)*sqrt(c*sin(a+b*x)))],

# Integrands of the form (a Sin[e+f x])^m (b Sec[e+f x])^n

# Integrands of the form Sin[e+f x]^m (b Sec[e+f x])^(n/2)

# n>0
[sin(e+f*x)^7*sqrt(b*sec(e+f*x)),x,3,2/13*b^7/(f*(b*sec(e+f*x))^(13/2))-2/3*b^5/(f*(b*sec(e+f*x))^(9/2))+6/5*b^3/(f*(b*sec(e+f*x))^(5/2))-2*b/(f*sqrt(b*sec(e+f*x)))],
[sin(e+f*x)^5*sqrt(b*sec(e+f*x)),x,3,-2/9*b^5/(f*(b*sec(e+f*x))^(9/2))+4/5*b^3/(f*(b*sec(e+f*x))^(5/2))-2*b/(f*sqrt(b*sec(e+f*x)))],
[sin(e+f*x)^3*sqrt(b*sec(e+f*x)),x,3,2/5*b^3/(f*(b*sec(e+f*x))^(5/2))-2*b/(f*sqrt(b*sec(e+f*x)))],
[sin(e+f*x)*sqrt(b*sec(e+f*x)),x,2,-2*b/(f*sqrt(b*sec(e+f*x)))],
[csc(e+f*x)*sqrt(b*sec(e+f*x)),x,5,arctan(sqrt(b*sec(e+f*x))/sqrt(b))*sqrt(b)/f-arctanh(sqrt(b*sec(e+f*x))/sqrt(b))*sqrt(b)/f],
[csc(e+f*x)^3*sqrt(b*sec(e+f*x)),x,6,-1/2*cot(e+f*x)^2*(b*sec(e+f*x))^(3/2)/(b*f)+3/4*arctan(sqrt(b*sec(e+f*x))/sqrt(b))*sqrt(b)/f-3/4*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))*sqrt(b)/f],
[csc(e+f*x)^5*sqrt(b*sec(e+f*x)),x,7,-7/16*cot(e+f*x)^2*(b*sec(e+f*x))^(3/2)/(b*f)-1/4*cot(e+f*x)^4*(b*sec(e+f*x))^(7/2)/(b^3*f)+21/32*arctan(sqrt(b*sec(e+f*x))/sqrt(b))*sqrt(b)/f-21/32*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))*sqrt(b)/f],
[sin(e+f*x)^6*sqrt(b*sec(e+f*x)),x,5,-40/77*b*sin(e+f*x)/(f*sqrt(b*sec(e+f*x)))-20/77*b*sin(e+f*x)^3/(f*sqrt(b*sec(e+f*x)))-2/11*b*sin(e+f*x)^5/(f*sqrt(b*sec(e+f*x)))+80/77*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[sin(e+f*x)^4*sqrt(b*sec(e+f*x)),x,4,-4/7*b*sin(e+f*x)/(f*sqrt(b*sec(e+f*x)))-2/7*b*sin(e+f*x)^3/(f*sqrt(b*sec(e+f*x)))+8/7*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[sin(e+f*x)^2*sqrt(b*sec(e+f*x)),x,3,-2/3*b*sin(e+f*x)/(f*sqrt(b*sec(e+f*x)))+4/3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[sqrt(b*sec(e+f*x)),x,2,2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)^2*sqrt(b*sec(e+f*x)),x,3,-b*csc(e+f*x)/(f*sqrt(b*sec(e+f*x)))+sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)^4*sqrt(b*sec(e+f*x)),x,4,-5/6*b*csc(e+f*x)/(f*sqrt(b*sec(e+f*x)))-1/3*b*csc(e+f*x)^3/(f*sqrt(b*sec(e+f*x)))+5/6*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)^6*sqrt(b*sec(e+f*x)),x,5,-3/4*b*csc(e+f*x)/(f*sqrt(b*sec(e+f*x)))-3/10*b*csc(e+f*x)^3/(f*sqrt(b*sec(e+f*x)))-1/5*b*csc(e+f*x)^5/(f*sqrt(b*sec(e+f*x)))+3/4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(3/2)*sin(e+f*x)^7,x,3,2/11*b^7/(f*(b*sec(e+f*x))^(11/2))-6/7*b^5/(f*(b*sec(e+f*x))^(7/2))+2*b^3/(f*(b*sec(e+f*x))^(3/2))+2*b*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(3/2)*sin(e+f*x)^5,x,3,-2/7*b^5/(f*(b*sec(e+f*x))^(7/2))+4/3*b^3/(f*(b*sec(e+f*x))^(3/2))+2*b*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(3/2)*sin(e+f*x)^3,x,3,2/3*b^3/(f*(b*sec(e+f*x))^(3/2))+2*b*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(3/2)*sin(e+f*x),x,2,2*b*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)*(b*sec(e+f*x))^(3/2),x,6,-b^(3/2)*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/f-b^(3/2)*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/f+2*b*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)^3*(b*sec(e+f*x))^(3/2),x,7,-5/4*b^(3/2)*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/f-5/4*b^(3/2)*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/f-1/2*cot(e+f*x)^2*(b*sec(e+f*x))^(5/2)/(b*f)+5/2*b*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(3/2)*sin(e+f*x)^6,x,5,8/3*b^3*sin(e+f*x)/(f*(b*sec(e+f*x))^(3/2))+20/9*b^3*sin(e+f*x)^3/(f*(b*sec(e+f*x))^(3/2))-16/3*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))+2*b*sin(e+f*x)^5*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(3/2)*sin(e+f*x)^4,x,4,12/5*b^3*sin(e+f*x)/(f*(b*sec(e+f*x))^(3/2))-24/5*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))+2*b*sin(e+f*x)^3*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(3/2)*sin(e+f*x)^2,x,3,-4*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))+2*b*sin(e+f*x)*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(3/2),x,3,-2*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))+2*b*sin(e+f*x)*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)^2*(b*sec(e+f*x))^(3/2),x,4,-3*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))-b*csc(e+f*x)*sqrt(b*sec(e+f*x))/f+3*b*sin(e+f*x)*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)^4*(b*sec(e+f*x))^(3/2),x,5,-7/2*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))-7/6*b*csc(e+f*x)*sqrt(b*sec(e+f*x))/f-1/3*b*csc(e+f*x)^3*sqrt(b*sec(e+f*x))/f+7/2*b*sin(e+f*x)*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(5/2)*sin(e+f*x)^7,x,3,2/9*b^7/(f*(b*sec(e+f*x))^(9/2))-6/5*b^5/(f*(b*sec(e+f*x))^(5/2))+2/3*b*(b*sec(e+f*x))^(3/2)/f+6*b^3/(f*sqrt(b*sec(e+f*x)))],
[(b*sec(e+f*x))^(5/2)*sin(e+f*x)^5,x,3,-2/5*b^5/(f*(b*sec(e+f*x))^(5/2))+2/3*b*(b*sec(e+f*x))^(3/2)/f+4*b^3/(f*sqrt(b*sec(e+f*x)))],
[(b*sec(e+f*x))^(5/2)*sin(e+f*x)^3,x,3,2/3*b*(b*sec(e+f*x))^(3/2)/f+2*b^3/(f*sqrt(b*sec(e+f*x)))],
[(b*sec(e+f*x))^(5/2)*sin(e+f*x),x,2,2/3*b*(b*sec(e+f*x))^(3/2)/f],
[csc(e+f*x)*(b*sec(e+f*x))^(5/2),x,6,b^(5/2)*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/f-b^(5/2)*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/f+2/3*b*(b*sec(e+f*x))^(3/2)/f],
[csc(e+f*x)^3*(b*sec(e+f*x))^(5/2),x,7,7/4*b^(5/2)*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/f-7/4*b^(5/2)*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/f+7/6*b*(b*sec(e+f*x))^(3/2)/f-1/2*cot(e+f*x)^2*(b*sec(e+f*x))^(7/2)/(b*f)],
[csc(e+f*x)^5*(b*sec(e+f*x))^(5/2),x,8,77/32*b^(5/2)*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/f-77/32*b^(5/2)*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/f+77/48*b*(b*sec(e+f*x))^(3/2)/f-11/16*cot(e+f*x)^2*(b*sec(e+f*x))^(7/2)/(b*f)-1/4*cot(e+f*x)^4*(b*sec(e+f*x))^(11/2)/(b^3*f)],
[(b*sec(e+f*x))^(5/2)*sin(e+f*x)^6,x,5,2/3*b*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^5/f+40/21*b^3*sin(e+f*x)/(f*sqrt(b*sec(e+f*x)))+20/21*b^3*sin(e+f*x)^3/(f*sqrt(b*sec(e+f*x)))-80/21*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(5/2)*sin(e+f*x)^4,x,4,2/3*b*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^3/f+4/3*b^3*sin(e+f*x)/(f*sqrt(b*sec(e+f*x)))-8/3*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(5/2)*sin(e+f*x)^2,x,3,2/3*b*(b*sec(e+f*x))^(3/2)*sin(e+f*x)/f-4/3*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[(b*sec(e+f*x))^(5/2),x,3,2/3*b*(b*sec(e+f*x))^(3/2)*sin(e+f*x)/f+2/3*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)^2*(b*sec(e+f*x))^(5/2),x,4,2/3*b*csc(e+f*x)*(b*sec(e+f*x))^(3/2)/f-5/3*b^3*csc(e+f*x)/(f*sqrt(b*sec(e+f*x)))+5/3*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],
[csc(e+f*x)^4*(b*sec(e+f*x))^(5/2),x,5,b*csc(e+f*x)*(b*sec(e+f*x))^(3/2)/f-1/3*b*csc(e+f*x)^3*(b*sec(e+f*x))^(3/2)/f-5/2*b^3*csc(e+f*x)/(f*sqrt(b*sec(e+f*x)))+5/2*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/f],

# n<0
[sin(e+f*x)^7/sqrt(b*sec(e+f*x)),x,3,2/15*b^7/(f*(b*sec(e+f*x))^(15/2))-6/11*b^5/(f*(b*sec(e+f*x))^(11/2))+6/7*b^3/(f*(b*sec(e+f*x))^(7/2))-2/3*b/(f*(b*sec(e+f*x))^(3/2))],
[sin(e+f*x)^5/sqrt(b*sec(e+f*x)),x,3,-2/11*b^5/(f*(b*sec(e+f*x))^(11/2))+4/7*b^3/(f*(b*sec(e+f*x))^(7/2))-2/3*b/(f*(b*sec(e+f*x))^(3/2))],
[sin(e+f*x)^3/sqrt(b*sec(e+f*x)),x,3,2/7*b^3/(f*(b*sec(e+f*x))^(7/2))-2/3*b/(f*(b*sec(e+f*x))^(3/2))],
[sin(e+f*x)/sqrt(b*sec(e+f*x)),x,2,-2/3*b/(f*(b*sec(e+f*x))^(3/2))],
[csc(e+f*x)/sqrt(b*sec(e+f*x)),x,5,-arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(f*sqrt(b))-arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(f*sqrt(b))],
[csc(e+f*x)^3/sqrt(b*sec(e+f*x)),x,6,-1/4*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(f*sqrt(b))-1/4*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(f*sqrt(b))-1/2*cot(e+f*x)^2*sqrt(b*sec(e+f*x))/(b*f)],
[csc(e+f*x)^5/sqrt(b*sec(e+f*x)),x,7,-1/4*cot(e+f*x)^4*(b*sec(e+f*x))^(5/2)/(b^3*f)-5/32*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(f*sqrt(b))-5/32*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(f*sqrt(b))-5/16*cot(e+f*x)^2*sqrt(b*sec(e+f*x))/(b*f)],
[sin(e+f*x)^6/sqrt(b*sec(e+f*x)),x,5,-8/39*b*sin(e+f*x)/(f*(b*sec(e+f*x))^(3/2))-20/117*b*sin(e+f*x)^3/(f*(b*sec(e+f*x))^(3/2))-2/13*b*sin(e+f*x)^5/(f*(b*sec(e+f*x))^(3/2))+16/39*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[sin(e+f*x)^4/sqrt(b*sec(e+f*x)),x,4,-4/15*b*sin(e+f*x)/(f*(b*sec(e+f*x))^(3/2))-2/9*b*sin(e+f*x)^3/(f*(b*sec(e+f*x))^(3/2))+8/15*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[sin(e+f*x)^2/sqrt(b*sec(e+f*x)),x,3,-2/5*b*sin(e+f*x)/(f*(b*sec(e+f*x))^(3/2))+4/5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[1/sqrt(b*sec(e+f*x)),x,2,2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[csc(e+f*x)^2/sqrt(b*sec(e+f*x)),x,3,-b*csc(e+f*x)/(f*(b*sec(e+f*x))^(3/2))-sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[csc(e+f*x)^4/sqrt(b*sec(e+f*x)),x,4,-1/2*b*csc(e+f*x)/(f*(b*sec(e+f*x))^(3/2))-1/3*b*csc(e+f*x)^3/(f*(b*sec(e+f*x))^(3/2))-1/2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[csc(e+f*x)^6/sqrt(b*sec(e+f*x)),x,5,-7/20*b*csc(e+f*x)/(f*(b*sec(e+f*x))^(3/2))-7/30*b*csc(e+f*x)^3/(f*(b*sec(e+f*x))^(3/2))-1/5*b*csc(e+f*x)^5/(f*(b*sec(e+f*x))^(3/2))-7/20*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[sin(e+f*x)^7/(b*sec(e+f*x))^(3/2),x,3,2/17*b^7/(f*(b*sec(e+f*x))^(17/2))-6/13*b^5/(f*(b*sec(e+f*x))^(13/2))+2/3*b^3/(f*(b*sec(e+f*x))^(9/2))-2/5*b/(f*(b*sec(e+f*x))^(5/2))],
[sin(e+f*x)^5/(b*sec(e+f*x))^(3/2),x,3,-2/13*b^5/(f*(b*sec(e+f*x))^(13/2))+4/9*b^3/(f*(b*sec(e+f*x))^(9/2))-2/5*b/(f*(b*sec(e+f*x))^(5/2))],
[sin(e+f*x)^3/(b*sec(e+f*x))^(3/2),x,3,2/9*b^3/(f*(b*sec(e+f*x))^(9/2))-2/5*b/(f*(b*sec(e+f*x))^(5/2))],
[sin(e+f*x)/(b*sec(e+f*x))^(3/2),x,2,-2/5*b/(f*(b*sec(e+f*x))^(5/2))],
[csc(e+f*x)/(b*sec(e+f*x))^(3/2),x,6,arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(3/2)*f)-arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(3/2)*f)+2/(b*f*sqrt(b*sec(e+f*x)))],
[csc(e+f*x)^3/(b*sec(e+f*x))^(3/2),x,6,-1/4*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(3/2)*f)+1/4*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(3/2)*f)-1/2*cot(e+f*x)^2*(b*sec(e+f*x))^(3/2)/(b^3*f)],
[csc(e+f*x)^5/(b*sec(e+f*x))^(3/2),x,7,-3/32*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(3/2)*f)+3/32*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(3/2)*f)-3/16*cot(e+f*x)^2*(b*sec(e+f*x))^(3/2)/(b^3*f)-1/4*cot(e+f*x)^4*(b*sec(e+f*x))^(3/2)/(b^3*f)],
[sin(e+f*x)^4/(b*sec(e+f*x))^(3/2),x,5,-12/77*b*sin(e+f*x)/(f*(b*sec(e+f*x))^(5/2))-2/11*b*sin(e+f*x)^3/(f*(b*sec(e+f*x))^(5/2))+8/77*sin(e+f*x)/(b*f*sqrt(b*sec(e+f*x)))+8/77*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^2*f)],
[sin(e+f*x)^2/(b*sec(e+f*x))^(3/2),x,4,-2/7*b*sin(e+f*x)/(f*(b*sec(e+f*x))^(5/2))+4/21*sin(e+f*x)/(b*f*sqrt(b*sec(e+f*x)))+4/21*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^2*f)],
[1/(b*sec(e+f*x))^(3/2),x,3,2/3*sin(e+f*x)/(b*f*sqrt(b*sec(e+f*x)))+2/3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^2*f)],
[csc(e+f*x)^2/(b*sec(e+f*x))^(3/2),x,3,-csc(e+f*x)/(b*f*sqrt(b*sec(e+f*x)))-sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^2*f)],
[csc(e+f*x)^4/(b*sec(e+f*x))^(3/2),x,4,1/6*csc(e+f*x)/(b*f*sqrt(b*sec(e+f*x)))-1/3*csc(e+f*x)^3/(b*f*sqrt(b*sec(e+f*x)))-1/6*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^2*f)],
[csc(e+f*x)^6/(b*sec(e+f*x))^(3/2),x,5,1/12*csc(e+f*x)/(b*f*sqrt(b*sec(e+f*x)))+1/30*csc(e+f*x)^3/(b*f*sqrt(b*sec(e+f*x)))-1/5*csc(e+f*x)^5/(b*f*sqrt(b*sec(e+f*x)))-1/12*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^2*f)],
[sin(e+f*x)^7/(b*sec(e+f*x))^(5/2),x,3,2/19*b^7/(f*(b*sec(e+f*x))^(19/2))-2/5*b^5/(f*(b*sec(e+f*x))^(15/2))+6/11*b^3/(f*(b*sec(e+f*x))^(11/2))-2/7*b/(f*(b*sec(e+f*x))^(7/2))],
[sin(e+f*x)^5/(b*sec(e+f*x))^(5/2),x,3,-2/15*b^5/(f*(b*sec(e+f*x))^(15/2))+4/11*b^3/(f*(b*sec(e+f*x))^(11/2))-2/7*b/(f*(b*sec(e+f*x))^(7/2))],
[sin(e+f*x)^3/(b*sec(e+f*x))^(5/2),x,3,2/11*b^3/(f*(b*sec(e+f*x))^(11/2))-2/7*b/(f*(b*sec(e+f*x))^(7/2))],
[sin(e+f*x)/(b*sec(e+f*x))^(5/2),x,2,-2/7*b/(f*(b*sec(e+f*x))^(7/2))],
[csc(e+f*x)/(b*sec(e+f*x))^(5/2),x,6,-arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(5/2)*f)-arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(5/2)*f)+2/3/(b*f*(b*sec(e+f*x))^(3/2))],
[csc(e+f*x)^3/(b*sec(e+f*x))^(5/2),x,6,3/4*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(5/2)*f)+3/4*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(5/2)*f)-1/2*cot(e+f*x)^2*sqrt(b*sec(e+f*x))/(b^3*f)],
[csc(e+f*x)^5/(b*sec(e+f*x))^(5/2),x,7,3/32*arctan(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(5/2)*f)+3/32*arctanh(sqrt(b*sec(e+f*x))/sqrt(b))/(b^(5/2)*f)-1/16*cot(e+f*x)^2*sqrt(b*sec(e+f*x))/(b^3*f)-1/4*cot(e+f*x)^4*sqrt(b*sec(e+f*x))/(b^3*f)],
[sin(e+f*x)^4/(b*sec(e+f*x))^(5/2),x,5,-4/39*b*sin(e+f*x)/(f*(b*sec(e+f*x))^(7/2))+8/195*sin(e+f*x)/(b*f*(b*sec(e+f*x))^(3/2))-2/13*b*sin(e+f*x)^3/(f*(b*sec(e+f*x))^(7/2))+8/65*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(b^2*f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[sin(e+f*x)^2/(b*sec(e+f*x))^(5/2),x,4,-2/9*b*sin(e+f*x)/(f*(b*sec(e+f*x))^(7/2))+4/45*sin(e+f*x)/(b*f*(b*sec(e+f*x))^(3/2))+4/15*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(b^2*f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[1/(b*sec(e+f*x))^(5/2),x,3,2/5*sin(e+f*x)/(b*f*(b*sec(e+f*x))^(3/2))+6/5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(b^2*f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[csc(e+f*x)^2/(b*sec(e+f*x))^(5/2),x,3,-csc(e+f*x)/(b*f*(b*sec(e+f*x))^(3/2))-3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(b^2*f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[csc(e+f*x)^4/(b*sec(e+f*x))^(5/2),x,4,1/2*csc(e+f*x)/(b*f*(b*sec(e+f*x))^(3/2))-1/3*csc(e+f*x)^3/(b*f*(b*sec(e+f*x))^(3/2))+1/2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(b^2*f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[csc(e+f*x)^6/(b*sec(e+f*x))^(5/2),x,5,3/20*csc(e+f*x)/(b*f*(b*sec(e+f*x))^(3/2))+1/10*csc(e+f*x)^3/(b*f*(b*sec(e+f*x))^(3/2))-1/5*csc(e+f*x)^5/(b*f*(b*sec(e+f*x))^(3/2))+3/20*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))/(b^2*f*sqrt(cos(e+f*x))*sqrt(b*sec(e+f*x)))],

# Integrands of the form (a Sin[e+f x])^(m/2) (b Sec[e+f x])^(n/2)

# n>0
[(a*sin(e+f*x))^(9/2)*sqrt(b*sec(e+f*x)),x,13,-7/16*a^3*b*(a*sin(e+f*x))^(3/2)/(f*sqrt(b*sec(e+f*x)))-1/4*a*b*(a*sin(e+f*x))^(7/2)/(f*sqrt(b*sec(e+f*x)))-21/32*a^(9/2)*arctan(1-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))+21/32*a^(9/2)*arctan(1+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))+21/64*a^(9/2)*log(sqrt(a)-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))-21/64*a^(9/2)*log(sqrt(a)+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))],
[(a*sin(e+f*x))^(5/2)*sqrt(b*sec(e+f*x)),x,12,-1/2*a*b*(a*sin(e+f*x))^(3/2)/(f*sqrt(b*sec(e+f*x)))-3/4*a^(5/2)*arctan(1-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))+3/4*a^(5/2)*arctan(1+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))+3/8*a^(5/2)*log(sqrt(a)-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))-3/8*a^(5/2)*log(sqrt(a)+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))],
[(a*sin(e+f*x))^(1/2)*sqrt(b*sec(e+f*x)),x,11,-arctan(1-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(a)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))+arctan(1+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(a)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))+1/2*log(sqrt(a)-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(a)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))-1/2*log(sqrt(a)+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(a)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(f*sqrt(2)*sqrt(b))],
[sqrt(b*sec(e+f*x))/(a*sin(e+f*x))^(3/2),x,1,-2*b/(a*f*sqrt(b*sec(e+f*x))*sqrt(a*sin(e+f*x)))],
[sqrt(b*sec(e+f*x))/(a*sin(e+f*x))^(7/2),x,2,-2/5*b/(a*f*(a*sin(e+f*x))^(5/2)*sqrt(b*sec(e+f*x)))-8/5*b/(a^3*f*sqrt(b*sec(e+f*x))*sqrt(a*sin(e+f*x)))],
[sqrt(b*sec(e+f*x))/(a*sin(e+f*x))^(11/2),x,3,-2/9*b/(a*f*(a*sin(e+f*x))^(9/2)*sqrt(b*sec(e+f*x)))-16/45*b/(a^3*f*(a*sin(e+f*x))^(5/2)*sqrt(b*sec(e+f*x)))-64/45*b/(a^5*f*sqrt(b*sec(e+f*x))*sqrt(a*sin(e+f*x)))],
[(a*sin(e+f*x))^(7/2)*sqrt(b*sec(e+f*x)),x,5,-1/3*a*b*(a*sin(e+f*x))^(5/2)/(f*sqrt(b*sec(e+f*x)))-5/6*a^3*b*sqrt(a*sin(e+f*x))/(f*sqrt(b*sec(e+f*x)))+5/12*a^4*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(f*sqrt(a*sin(e+f*x)))],
[(a*sin(e+f*x))^(3/2)*sqrt(b*sec(e+f*x)),x,4,-a*b*sqrt(a*sin(e+f*x))/(f*sqrt(b*sec(e+f*x)))+1/2*a^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(f*sqrt(a*sin(e+f*x)))],
[sqrt(b*sec(e+f*x))/(a*sin(e+f*x))^(1/2),x,3,sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(f*sqrt(a*sin(e+f*x)))],
[sqrt(b*sec(e+f*x))/(a*sin(e+f*x))^(5/2),x,4,-2/3*b/(a*f*(a*sin(e+f*x))^(3/2)*sqrt(b*sec(e+f*x)))+2/3*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(a^2*f*sqrt(a*sin(e+f*x)))],
[sqrt(b*sec(e+f*x))/(a*sin(e+f*x))^(9/2),x,5,-2/7*b/(a*f*(a*sin(e+f*x))^(7/2)*sqrt(b*sec(e+f*x)))-4/7*b/(a^3*f*(a*sin(e+f*x))^(3/2)*sqrt(b*sec(e+f*x)))+4/7*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(a^4*f*sqrt(a*sin(e+f*x)))],

# n<0
[sin(e+f*x)^(9/2)/sqrt(b*sec(e+f*x)),x,5,-7/30*b*sin(e+f*x)^(3/2)/(f*(b*sec(e+f*x))^(3/2))-1/5*b*sin(e+f*x)^(7/2)/(f*(b*sec(e+f*x))^(3/2))+7/20*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(e+f*x))/(f*(b*sec(e+f*x))^(1/2)*sqrt(sin(2*e+2*f*x)))],
[sin(e+f*x)^(5/2)/sqrt(b*sec(e+f*x)),x,4,-1/3*b*sin(e+f*x)^(3/2)/(f*(b*sec(e+f*x))^(3/2))+1/2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(e+f*x))/(f*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x)))],
[sin(e+f*x)^(1/2)/sqrt(b*sec(e+f*x)),x,3,sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(e+f*x))/(f*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x)))],
[1/(sin(e+f*x)^(3/2)*sqrt(b*sec(e+f*x))),x,4,-2*b/(f*(b*sec(e+f*x))^(3/2)*sqrt(sin(e+f*x)))-2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(e+f*x))/(f*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x)))],
[1/(sin(e+f*x)^(7/2)*sqrt(b*sec(e+f*x))),x,5,-2/5*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(5/2))-4/5*b/(f*(b*sec(e+f*x))^(3/2)*sqrt(sin(e+f*x)))-4/5*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(e+f*x))/(f*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x)))],
[sin(e+f*x)^(3/2)/sqrt(b*sec(e+f*x)),x,12,1/4*arctan(1-sqrt(2)*sqrt(b*cos(e+f*x))/(sqrt(b)*sqrt(sin(e+f*x))))*sqrt(b)/(f*sqrt(2)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x)))-1/4*arctan(1+sqrt(2)*sqrt(b*cos(e+f*x))/(sqrt(b)*sqrt(sin(e+f*x))))*sqrt(b)/(f*sqrt(2)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x)))-1/8*log(sqrt(b)+cot(e+f*x)*sqrt(b)-sqrt(2)*sqrt(b*cos(e+f*x))/sqrt(sin(e+f*x)))*sqrt(b)/(f*sqrt(2)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x)))+1/8*log(sqrt(b)+cot(e+f*x)*sqrt(b)+sqrt(2)*sqrt(b*cos(e+f*x))/sqrt(sin(e+f*x)))*sqrt(b)/(f*sqrt(2)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x)))-1/2*b*sqrt(sin(e+f*x))/(f*(b*sec(e+f*x))^(3/2))],
[1/(sin(e+f*x)^(1/2)*sqrt(b*sec(e+f*x))),x,11,arctan(1-sqrt(2)*sqrt(b*cos(e+f*x))/(sqrt(b)*sqrt(sin(e+f*x))))*sqrt(b)/(f*sqrt(2)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x)))-arctan(1+sqrt(2)*sqrt(b*cos(e+f*x))/(sqrt(b)*sqrt(sin(e+f*x))))*sqrt(b)/(f*sqrt(2)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x)))-1/2*log(sqrt(b)+cot(e+f*x)*sqrt(b)-sqrt(2)*sqrt(b*cos(e+f*x))/sqrt(sin(e+f*x)))*sqrt(b)/(f*sqrt(2)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x)))+1/2*log(sqrt(b)+cot(e+f*x)*sqrt(b)+sqrt(2)*sqrt(b*cos(e+f*x))/sqrt(sin(e+f*x)))*sqrt(b)/(f*sqrt(2)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x)))],
[1/(sin(e+f*x)^(5/2)*sqrt(b*sec(e+f*x))),x,1,-2/3*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(3/2))],
[1/(sin(e+f*x)^(9/2)*sqrt(b*sec(e+f*x))),x,2,-2/7*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(7/2))-8/21*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(3/2))],
[1/(sin(e+f*x)^(13/2)*sqrt(b*sec(e+f*x))),x,3,-2/11*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(11/2))-16/77*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(7/2))-64/231*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(3/2))],
[1/(sin(e+f*x)^(17/2)*sqrt(b*sec(e+f*x))),x,4,-2/15*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(15/2))-8/55*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(11/2))-64/385*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(7/2))-256/1155*b/(f*(b*sec(e+f*x))^(3/2)*sin(e+f*x)^(3/2))],
[(a*sin(e+f*x))^(9/2)/(b*sec(e+f*x))^(3/2),x,14,-7/192*a^3*(a*sin(e+f*x))^(3/2)/(b*f*sqrt(b*sec(e+f*x)))-1/48*a*(a*sin(e+f*x))^(7/2)/(b*f*sqrt(b*sec(e+f*x)))+1/6*(a*sin(e+f*x))^(11/2)/(a*b*f*sqrt(b*sec(e+f*x)))-7/128*a^(9/2)*arctan(1-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))+7/128*a^(9/2)*arctan(1+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))+7/256*a^(9/2)*log(sqrt(a)-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))-7/256*a^(9/2)*log(sqrt(a)+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))],
[(a*sin(e+f*x))^(5/2)/(b*sec(e+f*x))^(3/2),x,13,-1/16*a*(a*sin(e+f*x))^(3/2)/(b*f*sqrt(b*sec(e+f*x)))+1/4*(a*sin(e+f*x))^(7/2)/(a*b*f*sqrt(b*sec(e+f*x)))-3/32*a^(5/2)*arctan(1-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))+3/32*a^(5/2)*arctan(1+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))+3/64*a^(5/2)*log(sqrt(a)-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))-3/64*a^(5/2)*log(sqrt(a)+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))],
[(a*sin(e+f*x))^(1/2)/(b*sec(e+f*x))^(3/2),x,12,1/2*(a*sin(e+f*x))^(3/2)/(a*b*f*sqrt(b*sec(e+f*x)))-1/4*arctan(1-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(a)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))+1/4*arctan(1+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(a)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))+1/8*log(sqrt(a)-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(a)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))-1/8*log(sqrt(a)+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(a)*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(b^(5/2)*f*sqrt(2))],
[1/((b*sec(e+f*x))^(3/2)*(a*sin(e+f*x))^(3/2)),x,12,arctan(1-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(a^(3/2)*b^(5/2)*f*sqrt(2))-arctan(1+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/(sqrt(a)*sqrt(b*cos(e+f*x))))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(a^(3/2)*b^(5/2)*f*sqrt(2))-1/2*log(sqrt(a)-sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(a^(3/2)*b^(5/2)*f*sqrt(2))+1/2*log(sqrt(a)+sqrt(2)*sqrt(b)*sqrt(a*sin(e+f*x))/sqrt(b*cos(e+f*x))+sqrt(a)*tan(e+f*x))*sqrt(b*cos(e+f*x))*sqrt(b*sec(e+f*x))/(a^(3/2)*b^(5/2)*f*sqrt(2))+(-2)/(a*b*f*sqrt(b*sec(e+f*x))*sqrt(a*sin(e+f*x)))],
[1/((b*sec(e+f*x))^(3/2)*(a*sin(e+f*x))^(7/2)),x,1,-2/5*b/(a*f*(b*sec(e+f*x))^(5/2)*(a*sin(e+f*x))^(5/2))],
[(a*sin(e+f*x))^(7/2)/(b*sec(e+f*x))^(3/2),x,6,-1/30*a*(a*sin(e+f*x))^(5/2)/(b*f*sqrt(b*sec(e+f*x)))+1/5*(a*sin(e+f*x))^(9/2)/(a*b*f*sqrt(b*sec(e+f*x)))-1/12*a^3*sqrt(a*sin(e+f*x))/(b*f*sqrt(b*sec(e+f*x)))+1/24*a^4*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(b^2*f*sqrt(a*sin(e+f*x)))],
[(a*sin(e+f*x))^(3/2)/(b*sec(e+f*x))^(3/2),x,5,1/3*(a*sin(e+f*x))^(5/2)/(a*b*f*sqrt(b*sec(e+f*x)))-1/6*a*sqrt(a*sin(e+f*x))/(b*f*sqrt(b*sec(e+f*x)))+1/12*a^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(b^2*f*sqrt(a*sin(e+f*x)))],
[1/((b*sec(e+f*x))^(3/2)*(a*sin(e+f*x))^(1/2)),x,4,sqrt(a*sin(e+f*x))/(a*b*f*sqrt(b*sec(e+f*x)))+1/2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(b^2*f*sqrt(a*sin(e+f*x)))],
[1/((b*sec(e+f*x))^(3/2)*(a*sin(e+f*x))^(5/2)),x,4,(-2/3)/(a*b*f*(a*sin(e+f*x))^(3/2)*sqrt(b*sec(e+f*x)))-1/3*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(a^2*b^2*f*sqrt(a*sin(e+f*x)))],
[1/((b*sec(e+f*x))^(3/2)*(a*sin(e+f*x))^(9/2)),x,5,(-2/7)/(a*b*f*(a*sin(e+f*x))^(7/2)*sqrt(b*sec(e+f*x)))+2/21/(a^3*b*f*(a*sin(e+f*x))^(3/2)*sqrt(b*sec(e+f*x)))-2/21*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(a^4*b^2*f*sqrt(a*sin(e+f*x)))],
[1/((b*sec(e+f*x))^(3/2)*(a*sin(e+f*x))^(13/2)),x,6,(-2/11)/(a*b*f*(a*sin(e+f*x))^(11/2)*sqrt(b*sec(e+f*x)))+2/77/(a^3*b*f*(a*sin(e+f*x))^(7/2)*sqrt(b*sec(e+f*x)))+4/77/(a^5*b*f*(a*sin(e+f*x))^(3/2)*sqrt(b*sec(e+f*x)))-4/77*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(b*sec(e+f*x))*sqrt(sin(2*e+2*f*x))/(a^6*b^2*f*sqrt(a*sin(e+f*x)))],

# Integrands of the form (a Sin[e+f x])^(m/3) (b Sec[e+f x])^(n/3)

# Integrands of the form (a Sin[e+f x])^m (b Sec[e+f x])^n with m symbolic
[(d*sec(a+b*x))^(5/2)*(c*sin(a+b*x))^m,x,2,d*(cos(a+b*x)^2)^(3/4)*hypergeom([7/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(d*sec(a+b*x))^(3/2)*(c*sin(a+b*x))^(1+m)/(b*c*(1+m))],
[(d*sec(a+b*x))^(3/2)*(c*sin(a+b*x))^m,x,2,d*(cos(a+b*x)^2)^(1/4)*hypergeom([5/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)*sqrt(d*sec(a+b*x))/(b*c*(1+m))],
[(d*sec(a+b*x))^(1/2)*(c*sin(a+b*x))^m,x,2,(cos(a+b*x)^2)^(3/4)*hypergeom([3/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(d*sec(a+b*x))^(3/2)*(c*sin(a+b*x))^(1+m)/(b*c*d*(1+m))],
[(c*sin(a+b*x))^m/(d*sec(a+b*x))^(1/2),x,2,(cos(a+b*x)^2)^(1/4)*hypergeom([1/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)*sqrt(d*sec(a+b*x))/(b*c*d*(1+m))],
[(c*sin(a+b*x))^m/(d*sec(a+b*x))^(3/2),x,2,hypergeom([-1/4,1/2*(1+m)],[1/2*(3+m)],sin(a+b*x)^2)*(c*sin(a+b*x))^(1+m)/(b*c*d*(1+m)*(cos(a+b*x)^2)^(1/4)*sqrt(d*sec(a+b*x)))],

# Integrands of the form (a Sin[e+f x])^m (b Sec[e+f x])^n with n symbolic
[sec(e+f*x)^n*sin(e+f*x)^m,x,2,-hypergeom([1/2*(1-m),1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*sec(e+f*x)^(-1+n)*sin(e+f*x)^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/(f*(1-n))],
[sec(e+f*x)^n*(a*sin(e+f*x))^m,x,2,-a*hypergeom([1/2*(1-m),1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*sec(e+f*x)^(-1+n)*(a*sin(e+f*x))^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/(f*(1-n))],
[(b*sec(e+f*x))^n*sin(e+f*x)^m,x,2,-b*hypergeom([1/2*(1-m),1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*sec(e+f*x))^(-1+n)*sin(e+f*x)^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/(f*(1-n))],
[(b*sec(e+f*x))^n*(a*sin(e+f*x))^m,x,2,-a*b*hypergeom([1/2*(1-m),1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*sec(e+f*x))^(-1+n)*(a*sin(e+f*x))^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/(f*(1-n))],
[(b*sec(e+f*x))^n*sin(e+f*x)^5,x,3,-b^5*(b*sec(e+f*x))^(-5+n)/(f*(5-n))+2*b^3*(b*sec(e+f*x))^(-3+n)/(f*(3-n))-b*(b*sec(e+f*x))^(-1+n)/(f*(1-n))],
[(b*sec(e+f*x))^n*sin(e+f*x)^3,x,3,b^3*(b*sec(e+f*x))^(-3+n)/(f*(3-n))-b*(b*sec(e+f*x))^(-1+n)/(f*(1-n))],
[(b*sec(e+f*x))^n*sin(e+f*x),x,2,-b*(b*sec(e+f*x))^(-1+n)/(f*(1-n))],
[csc(e+f*x)*(b*sec(e+f*x))^n,x,2,-hypergeom([1,1/2*(1+n)],[1/2*(3+n)],sec(e+f*x)^2)*(b*sec(e+f*x))^(1+n)/(b*f*(1+n))],
[csc(e+f*x)^3*(b*sec(e+f*x))^n,x,2,hypergeom([2,1/2*(3+n)],[1/2*(5+n)],sec(e+f*x)^2)*(b*sec(e+f*x))^(3+n)/(b^3*f*(3+n))],
[(b*sec(e+f*x))^n*sin(e+f*x)^6,x,2,-b*hypergeom([-5/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*sec(e+f*x))^(-1+n)*sin(e+f*x)/(f*(1-n)*sqrt(sin(e+f*x)^2))],
[(b*sec(e+f*x))^n*sin(e+f*x)^4,x,2,-b*hypergeom([-3/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*sec(e+f*x))^(-1+n)*sin(e+f*x)/(f*(1-n)*sqrt(sin(e+f*x)^2))],
[(b*sec(e+f*x))^n*sin(e+f*x)^2,x,2,-b*hypergeom([-1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*sec(e+f*x))^(-1+n)*sin(e+f*x)/(f*(1-n)*sqrt(sin(e+f*x)^2))],
[(b*sec(e+f*x))^n,x,2,-b*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*sec(e+f*x))^(-1+n)*sin(e+f*x)/(f*(1-n)*sqrt(sin(e+f*x)^2))],
[csc(e+f*x)^2*(b*sec(e+f*x))^n,x,2,-b*csc(e+f*x)*hypergeom([3/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*sec(e+f*x))^(-1+n)*sqrt(sin(e+f*x)^2)/(f*(1-n))],
[csc(e+f*x)^4*(b*sec(e+f*x))^n,x,2,-b*csc(e+f*x)*hypergeom([5/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(b*sec(e+f*x))^(-1+n)*sqrt(sin(e+f*x)^2)/(f*(1-n))],
[(b*sec(a+b*x))^n*(c*sin(a+b*x))^(3/2),x,2,-c*hypergeom([-1/4,1/2*(1-n)],[1/2*(3-n)],cos(a+b*x)^2)*(b*sec(a+b*x))^(-1+n)*sqrt(c*sin(a+b*x))/((1-n)*(sin(a+b*x)^2)^(1/4))],
[(b*sec(a+b*x))^n*(c*sin(a+b*x))^(1/2),x,2,-c*hypergeom([1/4,1/2*(1-n)],[1/2*(3-n)],cos(a+b*x)^2)*(b*sec(a+b*x))^(-1+n)*(sin(a+b*x)^2)^(1/4)/((1-n)*sqrt(c*sin(a+b*x)))],
[(b*sec(a+b*x))^n/(c*sin(a+b*x))^(1/2),x,2,-c*hypergeom([3/4,1/2*(1-n)],[1/2*(3-n)],cos(a+b*x)^2)*(b*sec(a+b*x))^(-1+n)*(sin(a+b*x)^2)^(3/4)/((1-n)*(c*sin(a+b*x))^(3/2))],
[(b*sec(a+b*x))^n/(c*sin(a+b*x))^(3/2),x,2,-hypergeom([5/4,1/2*(1-n)],[1/2*(3-n)],cos(a+b*x)^2)*(b*sec(a+b*x))^(-1+n)*(sin(a+b*x)^2)^(1/4)/(c*(1-n)*sqrt(c*sin(a+b*x)))],

# Integrands of the form (a Sin[e+f x])^m (b Csc[e+f x])^n

# Integrands of the form Sin[e+f x]^m (b Csc[e+f x])^(n/2)

# n>0
[sin(e+f*x)^4*sqrt(d*csc(e+f*x)),x,5,-2/7*d^3*cos(e+f*x)/(f*(d*csc(e+f*x))^(5/2))-10/21*d*cos(e+f*x)/(f*sqrt(d*csc(e+f*x)))+10/21*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/f],
[sin(e+f*x)^3*sqrt(d*csc(e+f*x)),x,4,-2/5*d^2*cos(e+f*x)/(f*(d*csc(e+f*x))^(3/2))+6/5*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[sin(e+f*x)^2*sqrt(d*csc(e+f*x)),x,4,-2/3*d*cos(e+f*x)/(f*sqrt(d*csc(e+f*x)))+2/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/f],
[sin(e+f*x)*sqrt(d*csc(e+f*x)),x,3,2*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[sqrt(d*csc(e+f*x)),x,2,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/f],
[csc(e+f*x)*sqrt(d*csc(e+f*x)),x,4,-2*cos(e+f*x)*sqrt(d*csc(e+f*x))/f-2*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[csc(e+f*x)^2*sqrt(d*csc(e+f*x)),x,4,-2/3*cos(e+f*x)*(d*csc(e+f*x))^(3/2)/(d*f)+2/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/f],
[csc(e+f*x)^3*sqrt(d*csc(e+f*x)),x,5,-2/5*cos(e+f*x)*(d*csc(e+f*x))^(5/2)/(d^2*f)-6/5*cos(e+f*x)*sqrt(d*csc(e+f*x))/f-6/5*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[(d*csc(e+f*x))^(3/2)*sin(e+f*x)^5,x,5,-2/7*d^4*cos(e+f*x)/(f*(d*csc(e+f*x))^(5/2))-10/21*d^2*cos(e+f*x)/(f*sqrt(d*csc(e+f*x)))+10/21*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/f],
[(d*csc(e+f*x))^(3/2)*sin(e+f*x)^4,x,4,-2/5*d^3*cos(e+f*x)/(f*(d*csc(e+f*x))^(3/2))+6/5*d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[(d*csc(e+f*x))^(3/2)*sin(e+f*x)^3,x,4,-2/3*d^2*cos(e+f*x)/(f*sqrt(d*csc(e+f*x)))+2/3*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/f],
[(d*csc(e+f*x))^(3/2)*sin(e+f*x)^2,x,3,2*d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[(d*csc(e+f*x))^(3/2)*sin(e+f*x),x,3,2*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/f],
[(d*csc(e+f*x))^(3/2),x,3,-2*d*cos(e+f*x)*sqrt(d*csc(e+f*x))/f-2*d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[csc(e+f*x)*(d*csc(e+f*x))^(3/2),x,4,-2/3*cos(e+f*x)*(d*csc(e+f*x))^(3/2)/f+2/3*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/f],
[csc(e+f*x)^2*(d*csc(e+f*x))^(3/2),x,5,-2/5*cos(e+f*x)*(d*csc(e+f*x))^(5/2)/(d*f)-6/5*d*cos(e+f*x)*sqrt(d*csc(e+f*x))/f-6/5*d^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],

# n<0
[sin(e+f*x)^3/sqrt(d*csc(e+f*x)),x,5,-2/7*d^2*cos(e+f*x)/(f*(d*csc(e+f*x))^(5/2))-10/21*cos(e+f*x)/(f*sqrt(d*csc(e+f*x)))+10/21*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/(d*f)],
[sin(e+f*x)^2/sqrt(d*csc(e+f*x)),x,4,-2/5*d*cos(e+f*x)/(f*(d*csc(e+f*x))^(3/2))+6/5*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[sin(e+f*x)/sqrt(d*csc(e+f*x)),x,4,-2/3*cos(e+f*x)/(f*sqrt(d*csc(e+f*x)))+2/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/(d*f)],
[1/sqrt(d*csc(e+f*x)),x,2,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[csc(e+f*x)/sqrt(d*csc(e+f*x)),x,3,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/(d*f)],
[csc(e+f*x)^2/sqrt(d*csc(e+f*x)),x,4,-2*cos(e+f*x)*sqrt(d*csc(e+f*x))/(d*f)-2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[csc(e+f*x)^3/sqrt(d*csc(e+f*x)),x,4,-2/3*cos(e+f*x)*(d*csc(e+f*x))^(3/2)/(d^2*f)+2/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/(d*f)],
[sin(e+f*x)^2/(d*csc(e+f*x))^(3/2),x,5,-2/7*d*cos(e+f*x)/(f*(d*csc(e+f*x))^(5/2))-10/21*cos(e+f*x)/(d*f*sqrt(d*csc(e+f*x)))+10/21*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/(d^2*f)],
[sin(e+f*x)/(d*csc(e+f*x))^(3/2),x,4,-2/5*cos(e+f*x)/(f*(d*csc(e+f*x))^(3/2))+6/5*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(d*f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[1/(d*csc(e+f*x))^(3/2),x,3,-2/3*cos(e+f*x)/(d*f*sqrt(d*csc(e+f*x)))+2/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/(d^2*f)],
[csc(e+f*x)/(d*csc(e+f*x))^(3/2),x,3,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(d*f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[csc(e+f*x)^2/(d*csc(e+f*x))^(3/2),x,3,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/(d^2*f)],
[csc(e+f*x)^3/(d*csc(e+f*x))^(3/2),x,4,-2*cos(e+f*x)*sqrt(d*csc(e+f*x))/(d^2*f)-2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(d*f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],
[csc(e+f*x)^4/(d*csc(e+f*x))^(3/2),x,4,-2/3*cos(e+f*x)*(d*csc(e+f*x))^(3/2)/(d^3*f)+2/3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x))/(d^2*f)],
[csc(e+f*x)^5/(d*csc(e+f*x))^(3/2),x,5,-2/5*cos(e+f*x)*(d*csc(e+f*x))^(5/2)/(d^4*f)-6/5*cos(e+f*x)*sqrt(d*csc(e+f*x))/(d^2*f)-6/5*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2))/(d*f*sqrt(d*csc(e+f*x))*sqrt(sin(e+f*x)))],

# Integrands of the form (a Sin[e+f x])^(m/2) (b Csc[e+f x])^(n/2)

# Integrands of the form (a Sin[e+f x])^m (b Csc[e+f x])^n with m and n symbolic
[(b*csc(e+f*x))^n*(a*sin(e+f*x))^m,x,2,cos(e+f*x)*(b*csc(e+f*x))^n*hypergeom([1/2,1/2*(1+m-n)],[1/2*(3+m-n)],sin(e+f*x)^2)*(a*sin(e+f*x))^(1+m)/(a*f*(1+m-n)*sqrt(cos(e+f*x)^2))]]:
