# Maple integration test file: "4 Trig functions\4.1 Sine\4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).txt"

lst:=[

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

# Integrands of the form (a+a Sin[e+f x])^m (c Sin[e+f x])^n (A+B Sin[e+f x])
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x)),x,7,-a^3*(B*(27+14*n+2*n^2)+A*(28+15*n+2*n^2))*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(4+n)*(6+5*n+n^2))-a*B*cos(e+f*x)*(d*sin(e+f*x))^(1+n)*(a+a*sin(e+f*x))^2/(d*f*(4+n))-(A*(4+n)+B*(6+n))*cos(e+f*x)*(d*sin(e+f*x))^(1+n)*(a^3+a^3*sin(e+f*x))/(d*f*(3+n)*(4+n))+a^3*(B*(15+19*n+4*n^2)+A*(20+21*n+4*n^2))*cos(e+f*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+n)/(d*f*(4+n)*(2+3*n+n^2)*sqrt(cos(e+f*x)^2))+a^3*(B*(9+4*n)+A*(11+4*n))*cos(e+f*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+n)/(d^2*f*(2+n)*(3+n)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x)),x,6,-a^2*(A*(3+n)+B*(4+n))*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(2+n)*(3+n))-B*cos(e+f*x)*(d*sin(e+f*x))^(1+n)*(a^2+a^2*sin(e+f*x))/(d*f*(3+n))+a^2*(2*B*(1+n)+A*(3+2*n))*cos(e+f*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+n)/(d*f*(1+n)*(2+n)*sqrt(cos(e+f*x)^2))+a^2*(2*A*(3+n)+B*(5+2*n))*cos(e+f*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+n)/(d^2*f*(2+n)*(3+n)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))*(A+B*sin(e+f*x)),x,5,-a*B*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(2+n))+a*(B*(1+n)+A*(2+n))*cos(e+f*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+n)/(d*f*(1+n)*(2+n)*sqrt(cos(e+f*x)^2))+a*(A+B)*cos(e+f*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+n)/(d^2*f*(2+n)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^n*(A+B*sin(e+f*x))/(a+a*sin(e+f*x)),x,4,(A-B)*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(a+a*sin(e+f*x)))+(B-A*n+B*n)*cos(e+f*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+n)/(a*d*f*(1+n)*sqrt(cos(e+f*x)^2))+(A-B)*(1+n)*cos(e+f*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+n)/(a*d^2*f*(2+n)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^n*(A+B*sin(e+f*x))/(a+a*sin(e+f*x))^2,x,5,1/3*(B+2*A*(1-n)+2*B*n)*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(a^2*d*f*(1+sin(e+f*x)))+1/3*(A-B)*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(a+a*sin(e+f*x))^2)-1/3*n*(A-2*A*n+2*B*(1+n))*cos(e+f*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+n)/(a^2*d*f*(1+n)*sqrt(cos(e+f*x)^2))+1/3*(1+n)*(B+2*A*(1-n)+2*B*n)*cos(e+f*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+n)/(a^2*d^2*f*(2+n)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^n*(A+B*sin(e+f*x))/(a+a*sin(e+f*x))^3,x,6,1/5*(A-B)*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(a+a*sin(e+f*x))^3)+1/15*(A*(5-2*n)+2*B*n)*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(a*d*f*(a+a*sin(e+f*x))^2)+1/15*(1-n)*(7*A+3*B-4*A*n+4*B*n)*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(a^3+a^3*sin(e+f*x)))-1/15*n*(B*(3-n-4*n^2)+A*(2-9*n+4*n^2))*cos(e+f*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+n)/(a^3*d*f*(1+n)*sqrt(cos(e+f*x)^2))+1/15*(1-n)*(1+n)*(7*A+3*B-4*A*n+4*B*n)*cos(e+f*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+n)/(a^3*d^2*f*(2+n)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x)),x,6,-2*a*B*cos(e+f*x)*(d*sin(e+f*x))^(1+n)*(a+a*sin(e+f*x))^(3/2)/(d*f*(7+2*n))-2*a^3*(2*B*(115+203*n+104*n^2+16*n^3)+A*(301+478*n+224*n^2+32*n^3))*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))*(d*sin(e+f*x))^n/(f*(3+2*n)*(5+2*n)*(7+2*n)*sin(e+f*x)^n*sqrt(a+a*sin(e+f*x)))-2*a^3*(2*B*(35+23*n+4*n^2)+A*(77+50*n+8*n^2))*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(3+2*n)*(5+2*n)*(7+2*n)*sqrt(a+a*sin(e+f*x)))-2*a^2*(2*B*(5+n)+A*(7+2*n))*cos(e+f*x)*(d*sin(e+f*x))^(1+n)*sqrt(a+a*sin(e+f*x))/(d*f*(5+2*n)*(7+2*n))],
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x)),x,5,-2*a^2*(2*B*(9+13*n+4*n^2)+A*(25+30*n+8*n^2))*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))*(d*sin(e+f*x))^n/(f*(3+2*n)*(5+2*n)*sin(e+f*x)^n*sqrt(a+a*sin(e+f*x)))-2*a^2*(2*B*(3+n)+A*(5+2*n))*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(3+2*n)*(5+2*n)*sqrt(a+a*sin(e+f*x)))-2*a*B*cos(e+f*x)*(d*sin(e+f*x))^(1+n)*sqrt(a+a*sin(e+f*x))/(d*f*(5+2*n))],
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^(1/2)*(A+B*sin(e+f*x)),x,4,-2*a*(2*B*(1+n)+A*(3+2*n))*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))*(d*sin(e+f*x))^n/(f*(3+2*n)*sin(e+f*x)^n*sqrt(a+a*sin(e+f*x)))-2*a*B*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(3+2*n)*sqrt(a+a*sin(e+f*x)))],
[(d*sin(e+f*x))^n*(A+B*sin(e+f*x))/(a+a*sin(e+f*x))^(1/2),x,9,-(A-B)*AppellF1(1/2,-n,1,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)*(d*sin(e+f*x))^n/(f*sin(e+f*x)^n*sqrt(a+a*sin(e+f*x)))-2*B*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))*(d*sin(e+f*x))^n/(f*sin(e+f*x)^n*sqrt(a+a*sin(e+f*x)))],
[(d*sin(e+f*x))^n*(A+B*sin(e+f*x))/(a+a*sin(e+f*x))^(3/2),x,10,1/2*(A-B)*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(a+a*sin(e+f*x))^(3/2))-1/4*(A-4*A*n+B*(3+4*n))*AppellF1(1/2,-n,1,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)*(d*sin(e+f*x))^n/(a*f*sin(e+f*x)^n*sqrt(a+a*sin(e+f*x)))-1/2*(A-B)*(1+2*n)*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))*(d*sin(e+f*x))^n/(a*f*sin(e+f*x)^n*sqrt(a+a*sin(e+f*x)))],
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,9,-2^(3/2+m)*B*AppellF1(1/2,-n,-1/2-m,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)*(d*sin(e+f*x))^n*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(f*sin(e+f*x)^n)-2^(1/2+m)*(A-B)*AppellF1(1/2,-n,1/2-m,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)*(d*sin(e+f*x))^n*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(f*sin(e+f*x)^n)],
[(d*sin(e+f*x))^n*(a-a*sin(e+f*x))*(a+a*sin(e+f*x))^m,x,4,AppellF1(1+n,-1/2,1/2-m,2+n,sin(e+f*x),-sin(e+f*x))*sec(e+f*x)*(d*sin(e+f*x))^(1+n)*(1+sin(e+f*x))^(1/2-m)*(a-a*sin(e+f*x))*(a+a*sin(e+f*x))^m/(d*f*(1+n)*sqrt(1-sin(e+f*x)))],
[sin(c+d*x)^n*(a+a*sin(c+d*x))^(-2-n)*(-1-n-(-2-n)*sin(c+d*x)),x,1,-cos(c+d*x)*sin(c+d*x)^(1+n)*(a+a*sin(c+d*x))^(-2-n)/d],
[sin(c+d*x)^(-2-m)*(a+a*sin(c+d*x))^m*(1+m-m*sin(c+d*x)),x,1,-cos(c+d*x)*sin(c+d*x)^(-1-m)*(a+a*sin(c+d*x))^m/d],

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

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

# m>0

# m<0
[sin(e+f*x)^2*(A+B*sin(e+f*x))/(a+b*sin(e+f*x))^2,x,6,(A*b-2*a*B)*x/b^3-2*a*(a^2*A*b-2*A*b^3-2*a^3*B+3*a*b^2*B)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^3*(a^2-b^2)^(3/2)*f)-B*cos(e+f*x)/(b^2*f)+a^2*(A*b-a*B)*cos(e+f*x)/(b^2*(a^2-b^2)*f*(a+b*sin(e+f*x)))],

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

# Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x])

# Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x])

# m>0
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^4,x,7,7/16*a*(2*A-B)*c^4*x+7/24*a*(2*A-B)*c^4*cos(e+f*x)^3/f+7/16*a*(2*A-B)*c^4*cos(e+f*x)*sin(e+f*x)/f-1/6*a*B*c*cos(e+f*x)^3*(c-c*sin(e+f*x))^3/f+1/10*a*(2*A-B)*cos(e+f*x)^3*(c^2-c^2*sin(e+f*x))^2/f+7/40*a*(2*A-B)*cos(e+f*x)^3*(c^4-c^4*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^3,x,6,1/8*a*(5*A-2*B)*c^3*x+1/12*a*(5*A-2*B)*c^3*cos(e+f*x)^3/f+1/8*a*(5*A-2*B)*c^3*cos(e+f*x)*sin(e+f*x)/f-1/5*a*B*c*cos(e+f*x)^3*(c-c*sin(e+f*x))^2/f+1/20*a*(5*A-2*B)*cos(e+f*x)^3*(c^3-c^3*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^2,x,5,1/8*a*(4*A-B)*c^2*x+1/3*a*(A-B)*c^2*cos(e+f*x)^3/f+1/8*a*(4*A-B)*c^2*cos(e+f*x)*sin(e+f*x)/f+1/4*a*B*c^2*cos(e+f*x)^3*sin(e+f*x)/f,1/8*a*(4*A-B)*c^2*x+1/12*a*(4*A-B)*c^2*cos(e+f*x)^3/f+1/8*a*(4*A-B)*c^2*cos(e+f*x)*sin(e+f*x)/f-1/4*a*B*cos(e+f*x)^3*(c^2-c^2*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c-c*sin(e+f*x)),x,4,1/2*a*A*c*x-1/3*a*B*c*cos(e+f*x)^3/f+1/2*a*A*c*cos(e+f*x)*sin(e+f*x)/f],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x)),x,4,-a*(A+2*B)*x/c+a*B*cos(e+f*x)/(c*f)+2*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^2,x,4,a*B*x/c^2-1/3*a*(A+7*B)*cos(e+f*x)/(c^2*f*(1-sin(e+f*x)))+2/3*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^2)],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^3,x,4,2/5*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^3)-1/15*a*(A+11*B)*c*cos(e+f*x)/(f*(c^2-c^2*sin(e+f*x))^2)-1/15*a*(A-4*B)*cos(e+f*x)/(f*(c^3-c^3*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^4,x,5,2/7*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^4)-1/35*a*(A+15*B)*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^3)-1/105*a*(2*A-5*B)*cos(e+f*x)/(f*(c^2-c^2*sin(e+f*x))^2)-1/105*a*(2*A-5*B)*cos(e+f*x)/(f*(c^4-c^4*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^5,x,6,2/9*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^5)-1/63*a*(A+19*B)*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^4)-1/105*a*(A-2*B)*c*cos(e+f*x)/(f*(c^2-c^2*sin(e+f*x))^3)-2/315*a*(A-2*B)*c*cos(e+f*x)/(f*(c^3-c^3*sin(e+f*x))^2)-2/315*a*(A-2*B)*cos(e+f*x)/(f*(c^5-c^5*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^5,x,8,9/128*a^2*(8*A-3*B)*c^5*x+3/80*a^2*(8*A-3*B)*c^5*cos(e+f*x)^5/f+9/128*a^2*(8*A-3*B)*c^5*cos(e+f*x)*sin(e+f*x)/f+3/64*a^2*(8*A-3*B)*c^5*cos(e+f*x)^3*sin(e+f*x)/f+1/56*a^2*(8*A-3*B)*c^3*cos(e+f*x)^5*(c-c*sin(e+f*x))^2/f-1/8*a^2*B*c^2*cos(e+f*x)^5*(c-c*sin(e+f*x))^3/f+3/112*a^2*(8*A-3*B)*cos(e+f*x)^5*(c^5-c^5*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^4,x,7,1/16*a^2*(7*A-2*B)*c^4*x+1/30*a^2*(7*A-2*B)*c^4*cos(e+f*x)^5/f+1/16*a^2*(7*A-2*B)*c^4*cos(e+f*x)*sin(e+f*x)/f+1/24*a^2*(7*A-2*B)*c^4*cos(e+f*x)^3*sin(e+f*x)/f-1/7*a^2*B*cos(e+f*x)^5*(c^2-c^2*sin(e+f*x))^2/f+1/42*a^2*(7*A-2*B)*cos(e+f*x)^5*(c^4-c^4*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^3,x,6,1/16*a^2*(6*A-B)*c^3*x+1/30*a^2*(6*A-B)*c^3*cos(e+f*x)^5/f+1/16*a^2*(6*A-B)*c^3*cos(e+f*x)*sin(e+f*x)/f+1/24*a^2*(6*A-B)*c^3*cos(e+f*x)^3*sin(e+f*x)/f-1/6*a^2*B*cos(e+f*x)^5*(c^3-c^3*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^2,x,5,3/8*a^2*A*c^2*x-1/5*a^2*B*c^2*cos(e+f*x)^5/f+3/8*a^2*A*c^2*cos(e+f*x)*sin(e+f*x)/f+1/4*a^2*A*c^2*cos(e+f*x)^3*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x)),x,5,1/8*a^2*(4*A+B)*c*x-1/12*a^2*(4*A+B)*c*cos(e+f*x)^3/f+1/8*a^2*(4*A+B)*c*cos(e+f*x)*sin(e+f*x)/f-1/4*B*c*cos(e+f*x)^3*(a^2+a^2*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x)),x,5,-3/2*a^2*(2*A+3*B)*x/c+3/2*a^2*(2*A+3*B)*cos(e+f*x)/(c*f)+a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^3)+1/2*a^2*(2*A+3*B)*cos(e+f*x)^3/(f*(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^2,x,5,a^2*(A+4*B)*x/c^2-a^2*(A+4*B)*cos(e+f*x)/(c^2*f)+1/3*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^4)-2/3*a^2*(A+4*B)*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^2)],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^3,x,5,-a^2*B*x/c^3+1/5*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^5)-2/3*a^2*B*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^3)+2*a^2*B*cos(e+f*x)/(f*(c^3-c^3*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^4,x,3,1/7*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^6)+1/35*a^2*(A-6*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^5)],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^5,x,4,1/9*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^7)+1/63*a^2*(2*A-7*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^6)+1/315*a^2*(2*A-7*B)*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^5)],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^6,x,5,1/11*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^8)+1/99*a^2*(3*A-8*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^7)+2/693*a^2*(3*A-8*B)*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^6)+2/3465*a^2*(3*A-8*B)*cos(e+f*x)^5/(c*f*(c-c*sin(e+f*x))^5)],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^7,x,6,1/13*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^9)+1/143*a^2*(4*A-9*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^8)+1/429*a^2*(4*A-9*B)*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^7)+2/3003*a^2*(4*A-9*B)*cos(e+f*x)^5/(c*f*(c-c*sin(e+f*x))^6)+2/15015*a^2*(4*A-9*B)*cos(e+f*x)^5/(c^2*f*(c-c*sin(e+f*x))^5)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^6,x,9,11/256*a^3*(10*A-3*B)*c^6*x+11/560*a^3*(10*A-3*B)*c^6*cos(e+f*x)^7/f+11/256*a^3*(10*A-3*B)*c^6*cos(e+f*x)*sin(e+f*x)/f+11/384*a^3*(10*A-3*B)*c^6*cos(e+f*x)^3*sin(e+f*x)/f+11/480*a^3*(10*A-3*B)*c^6*cos(e+f*x)^5*sin(e+f*x)/f-1/10*a^3*B*cos(e+f*x)^7*(c^2-c^2*sin(e+f*x))^3/f+1/90*a^3*(10*A-3*B)*cos(e+f*x)^7*(c^3-c^3*sin(e+f*x))^2/f+11/720*a^3*(10*A-3*B)*cos(e+f*x)^7*(c^6-c^6*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^5,x,8,5/128*a^3*(9*A-2*B)*c^5*x+1/56*a^3*(9*A-2*B)*c^5*cos(e+f*x)^7/f+5/128*a^3*(9*A-2*B)*c^5*cos(e+f*x)*sin(e+f*x)/f+5/192*a^3*(9*A-2*B)*c^5*cos(e+f*x)^3*sin(e+f*x)/f+1/48*a^3*(9*A-2*B)*c^5*cos(e+f*x)^5*sin(e+f*x)/f-1/9*a^3*B*c^3*cos(e+f*x)^7*(c-c*sin(e+f*x))^2/f+1/72*a^3*(9*A-2*B)*cos(e+f*x)^7*(c^5-c^5*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^4,x,7,5/128*a^3*(8*A-B)*c^4*x+1/56*a^3*(8*A-B)*c^4*cos(e+f*x)^7/f+5/128*a^3*(8*A-B)*c^4*cos(e+f*x)*sin(e+f*x)/f+5/192*a^3*(8*A-B)*c^4*cos(e+f*x)^3*sin(e+f*x)/f+1/48*a^3*(8*A-B)*c^4*cos(e+f*x)^5*sin(e+f*x)/f-1/8*a^3*B*cos(e+f*x)^7*(c^4-c^4*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^3,x,6,5/16*a^3*A*c^3*x-1/7*a^3*B*c^3*cos(e+f*x)^7/f+5/16*a^3*A*c^3*cos(e+f*x)*sin(e+f*x)/f+5/24*a^3*A*c^3*cos(e+f*x)^3*sin(e+f*x)/f+1/6*a^3*A*c^3*cos(e+f*x)^5*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^2,x,6,1/16*a^3*(6*A+B)*c^2*x-1/30*a^3*(6*A+B)*c^2*cos(e+f*x)^5/f+1/16*a^3*(6*A+B)*c^2*cos(e+f*x)*sin(e+f*x)/f+1/24*a^3*(6*A+B)*c^2*cos(e+f*x)^3*sin(e+f*x)/f-1/6*B*c^2*cos(e+f*x)^5*(a^3+a^3*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x)),x,6,1/8*a^3*(5*A+2*B)*c*x-1/12*a^3*(5*A+2*B)*c*cos(e+f*x)^3/f+1/8*a^3*(5*A+2*B)*c*cos(e+f*x)*sin(e+f*x)/f-1/5*a*B*c*cos(e+f*x)^3*(a+a*sin(e+f*x))^2/f-1/20*(5*A+2*B)*c*cos(e+f*x)^3*(a^3+a^3*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x)),x,6,-5/2*a^3*(3*A+4*B)*x/c+5/3*a^3*(3*A+4*B)*cos(e+f*x)^3/(c*f)-5/2*a^3*(3*A+4*B)*cos(e+f*x)*sin(e+f*x)/(c*f)+a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^4)+2*a^3*(3*A+4*B)*c^3*cos(e+f*x)^5/(f*(c^2-c^2*sin(e+f*x))^2)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^2,x,6,5/2*a^3*(2*A+5*B)*x/c^2-5/2*a^3*(2*A+5*B)*cos(e+f*x)/(c^2*f)+1/3*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^5)-2/3*a^3*(2*A+5*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^3)-5/6*a^3*(2*A+5*B)*cos(e+f*x)^3/(f*(c^2-c^2*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^3,x,6,-a^3*(A+6*B)*x/c^3+a^3*(A+6*B)*cos(e+f*x)/(c^3*f)+1/5*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^6)-2/15*a^3*(A+6*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^4)+2/3*a^3*(A+6*B)*c^3*cos(e+f*x)^3/(f*(c^3-c^3*sin(e+f*x))^2)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^4,x,6,a^3*B*x/c^4+1/7*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^7)-2/5*a^3*B*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^5)+2/3*a^3*B*c^2*cos(e+f*x)^3/(f*(c^2-c^2*sin(e+f*x))^3)-2*a^3*B*cos(e+f*x)/(f*(c^4-c^4*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^5,x,3,1/9*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^8)+1/63*a^3*(A-8*B)*c^2*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^7)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^6,x,4,1/11*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^9)+1/99*a^3*(2*A-9*B)*c^2*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^8)+1/693*a^3*(2*A-9*B)*c*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^7)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^7,x,5,1/13*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^10)+1/143*a^3*(3*A-10*B)*c^2*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^9)+2/1287*a^3*(3*A-10*B)*c*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^8)+2/9009*a^3*(3*A-10*B)*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^7)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^8,x,6,1/15*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^11)+1/195*a^3*(4*A-11*B)*c^2*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^10)+1/715*a^3*(4*A-11*B)*c*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^9)+2/6435*a^3*(4*A-11*B)*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^8)+2/45045*a^3*(4*A-11*B)*cos(e+f*x)^7/(c*f*(c-c*sin(e+f*x))^7)],

# m<0
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^4/(a+a*sin(e+f*x)),x,7,-35/8*(4*A-5*B)*c^4*x/a-35/12*(4*A-5*B)*c^4*cos(e+f*x)^3/(a*f)-35/8*(4*A-5*B)*c^4*cos(e+f*x)*sin(e+f*x)/(a*f)-a^4*(A-B)*c^4*cos(e+f*x)^9/(f*(a+a*sin(e+f*x))^5)-2*a^2*(4*A-5*B)*c^4*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^3)-7/4*(4*A-5*B)*c^4*cos(e+f*x)^5/(f*(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^3/(a+a*sin(e+f*x)),x,6,-5/2*(3*A-4*B)*c^3*x/a-5/3*(3*A-4*B)*c^3*cos(e+f*x)^3/(a*f)-5/2*(3*A-4*B)*c^3*cos(e+f*x)*sin(e+f*x)/(a*f)-a^3*(A-B)*c^3*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^4)-2*a^3*(3*A-4*B)*c^3*cos(e+f*x)^5/(f*(a^2+a^2*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^2/(a+a*sin(e+f*x)),x,5,-3/2*(2*A-3*B)*c^2*x/a-3/2*(2*A-3*B)*c^2*cos(e+f*x)/(a*f)-a^2*(A-B)*c^2*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^3)-1/2*(2*A-3*B)*c^2*cos(e+f*x)^3/(f*(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))/(a+a*sin(e+f*x)),x,4,-(A-2*B)*c*x/a+B*c*cos(e+f*x)/(a*f)-2*(A-B)*c*cos(e+f*x)/(f*(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))),x,4,B*sec(e+f*x)/(a*c*f)+A*tan(e+f*x)/(a*c*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^2),x,4,1/3*(A+B)*sec(e+f*x)/(a*f*(c^2-c^2*sin(e+f*x)))+1/3*(2*A-B)*tan(e+f*x)/(a*c^2*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^3),x,5,1/5*(A+B)*sec(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^2)+1/15*(3*A-2*B)*sec(e+f*x)/(a*f*(c^3-c^3*sin(e+f*x)))+2/15*(3*A-2*B)*tan(e+f*x)/(a*c^3*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^4),x,6,1/7*(A+B)*sec(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^3)+1/35*(4*A-3*B)*sec(e+f*x)/(a*f*(c^2-c^2*sin(e+f*x))^2)+1/35*(4*A-3*B)*sec(e+f*x)/(a*f*(c^4-c^4*sin(e+f*x)))+2/35*(4*A-3*B)*tan(e+f*x)/(a*c^4*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^5/(a+a*sin(e+f*x))^2,x,8,105/8*(4*A-7*B)*c^5*x/a^2+35/4*(4*A-7*B)*c^5*cos(e+f*x)^3/(a^2*f)+105/8*(4*A-7*B)*c^5*cos(e+f*x)*sin(e+f*x)/(a^2*f)-1/3*a^5*(A-B)*c^5*cos(e+f*x)^11/(f*(a+a*sin(e+f*x))^7)+2/3*a^3*(4*A-7*B)*c^5*cos(e+f*x)^9/(f*(a+a*sin(e+f*x))^5)+6*a^4*(4*A-7*B)*c^5*cos(e+f*x)^7/(f*(a^2+a^2*sin(e+f*x))^3)+21/4*(4*A-7*B)*c^5*cos(e+f*x)^5/(f*(a^2+a^2*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^4/(a+a*sin(e+f*x))^2,x,7,35/2*(A-2*B)*c^4*x/a^2+35/3*(A-2*B)*c^4*cos(e+f*x)^3/(a^2*f)+35/2*(A-2*B)*c^4*cos(e+f*x)*sin(e+f*x)/(a^2*f)-1/3*a^4*(A-B)*c^4*cos(e+f*x)^9/(f*(a+a*sin(e+f*x))^6)+2*a^2*(A-2*B)*c^4*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^4)+14*(A-2*B)*c^4*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^3/(a+a*sin(e+f*x))^2,x,6,5/2*(2*A-5*B)*c^3*x/a^2+5/2*(2*A-5*B)*c^3*cos(e+f*x)/(a^2*f)-1/3*a^3*(A-B)*c^3*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^5)+2/3*a*(2*A-5*B)*c^3*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^3)+5/6*(2*A-5*B)*c^3*cos(e+f*x)^3/(f*(a^2+a^2*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^2/(a+a*sin(e+f*x))^2,x,5,(A-4*B)*c^2*x/a^2+(A-4*B)*c^2*cos(e+f*x)/(a^2*f)-1/3*a^2*(A-B)*c^2*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^4)+2/3*(A-4*B)*c^2*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))/(a+a*sin(e+f*x))^2,x,4,-B*c*x/a^2+1/3*(A-7*B)*c*cos(e+f*x)/(a^2*f*(1+sin(e+f*x)))-2/3*(A-B)*c*cos(e+f*x)/(f*(a+a*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))),x,4,-1/3*(A-B)*sec(e+f*x)/(c*f*(a^2+a^2*sin(e+f*x)))+1/3*(2*A+B)*tan(e+f*x)/(a^2*c*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^2),x,4,1/3*B*sec(e+f*x)^3/(a^2*c^2*f)+A*tan(e+f*x)/(a^2*c^2*f)+1/3*A*tan(e+f*x)^3/(a^2*c^2*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^3),x,4,1/5*(A+B)*sec(e+f*x)^3/(a^2*f*(c^3-c^3*sin(e+f*x)))+1/5*(4*A-B)*tan(e+f*x)/(a^2*c^3*f)+1/15*(4*A-B)*tan(e+f*x)^3/(a^2*c^3*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^4),x,5,1/7*(A+B)*sec(e+f*x)^3/(a^2*f*(c^2-c^2*sin(e+f*x))^2)+1/35*(5*A-2*B)*sec(e+f*x)^3/(a^2*f*(c^4-c^4*sin(e+f*x)))+4/35*(5*A-2*B)*tan(e+f*x)/(a^2*c^4*f)+4/105*(5*A-2*B)*tan(e+f*x)^3/(a^2*c^4*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^5),x,6,1/9*(A+B)*sec(e+f*x)^3/(a^2*c^2*f*(c-c*sin(e+f*x))^3)+1/21*(2*A-B)*sec(e+f*x)^3/(a^2*c^3*f*(c-c*sin(e+f*x))^2)+1/21*(2*A-B)*sec(e+f*x)^3/(a^2*f*(c^5-c^5*sin(e+f*x)))+4/21*(2*A-B)*tan(e+f*x)/(a^2*c^5*f)+4/63*(2*A-B)*tan(e+f*x)^3/(a^2*c^5*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^5/(a+a*sin(e+f*x))^3,x,8,-21/2*(3*A-8*B)*c^5*x/a^3-7*(3*A-8*B)*c^5*cos(e+f*x)^3/(a^3*f)-21/2*(3*A-8*B)*c^5*cos(e+f*x)*sin(e+f*x)/(a^3*f)-1/5*a^5*(A-B)*c^5*cos(e+f*x)^11/(f*(a+a*sin(e+f*x))^8)+2/15*a^3*(3*A-8*B)*c^5*cos(e+f*x)^9/(f*(a+a*sin(e+f*x))^6)-6/5*a^5*(3*A-8*B)*c^5*cos(e+f*x)^7/(f*(a^2+a^2*sin(e+f*x))^4)-42/5*a^5*(3*A-8*B)*c^5*cos(e+f*x)^5/(f*(a^4+a^4*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^4/(a+a*sin(e+f*x))^3,x,7,-7/2*(2*A-7*B)*c^4*x/a^3-7/2*(2*A-7*B)*c^4*cos(e+f*x)/(a^3*f)-1/5*a^4*(A-B)*c^4*cos(e+f*x)^9/(f*(a+a*sin(e+f*x))^7)+2/15*a^2*(2*A-7*B)*c^4*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^5)-14/15*(2*A-7*B)*c^4*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^3)-7/6*(2*A-7*B)*c^4*cos(e+f*x)^3/(f*(a^3+a^3*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^3/(a+a*sin(e+f*x))^3,x,6,-(A-6*B)*c^3*x/a^3-(A-6*B)*c^3*cos(e+f*x)/(a^3*f)-1/5*a^3*(A-B)*c^3*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^6)+2/15*a*(A-6*B)*c^3*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^4)-2/3*a^3*(A-6*B)*c^3*cos(e+f*x)^3/(f*(a^3+a^3*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^2/(a+a*sin(e+f*x))^3,x,5,B*c^2*x/a^3-1/5*a^2*(A-B)*c^2*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^5)-2/3*B*c^2*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^3)+2*B*c^2*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))/(a+a*sin(e+f*x))^3,x,4,-2/5*(A-B)*c*cos(e+f*x)/(f*(a+a*sin(e+f*x))^3)+1/15*a*(A-11*B)*c*cos(e+f*x)/(f*(a^2+a^2*sin(e+f*x))^2)+1/15*(A+4*B)*c*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))),x,5,-1/5*(A-B)*sec(e+f*x)/(a*c*f*(a+a*sin(e+f*x))^2)-1/15*(3*A+2*B)*sec(e+f*x)/(c*f*(a^3+a^3*sin(e+f*x)))+2/15*(3*A+2*B)*tan(e+f*x)/(a^3*c*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^2),x,4,-1/5*(A-B)*sec(e+f*x)^3/(c^2*f*(a^3+a^3*sin(e+f*x)))+1/5*(4*A+B)*tan(e+f*x)/(a^3*c^2*f)+1/15*(4*A+B)*tan(e+f*x)^3/(a^3*c^2*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^3),x,4,1/5*B*sec(e+f*x)^5/(a^3*c^3*f)+A*tan(e+f*x)/(a^3*c^3*f)+2/3*A*tan(e+f*x)^3/(a^3*c^3*f)+1/5*A*tan(e+f*x)^5/(a^3*c^3*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^4),x,4,1/7*(A+B)*sec(e+f*x)^5/(a^3*f*(c^4-c^4*sin(e+f*x)))+1/7*(6*A-B)*tan(e+f*x)/(a^3*c^4*f)+2/21*(6*A-B)*tan(e+f*x)^3/(a^3*c^4*f)+1/35*(6*A-B)*tan(e+f*x)^5/(a^3*c^4*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^5),x,5,1/9*(A+B)*sec(e+f*x)^5/(a^3*c^3*f*(c-c*sin(e+f*x))^2)+1/63*(7*A-2*B)*sec(e+f*x)^5/(a^3*f*(c^5-c^5*sin(e+f*x)))+2/21*(7*A-2*B)*tan(e+f*x)/(a^3*c^5*f)+4/63*(7*A-2*B)*tan(e+f*x)^3/(a^3*c^5*f)+2/105*(7*A-2*B)*tan(e+f*x)^5/(a^3*c^5*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^6),x,6,1/11*(A+B)*sec(e+f*x)^5/(a^3*f*(c^2-c^2*sin(e+f*x))^3)+1/99*(8*A-3*B)*sec(e+f*x)^5/(a^3*f*(c^3-c^3*sin(e+f*x))^2)+1/99*(8*A-3*B)*sec(e+f*x)^5/(a^3*f*(c^6-c^6*sin(e+f*x)))+2/33*(8*A-3*B)*tan(e+f*x)/(a^3*c^6*f)+4/99*(8*A-3*B)*tan(e+f*x)^3/(a^3*c^6*f)+2/165*(8*A-3*B)*tan(e+f*x)^5/(a^3*c^6*f)],

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

# m>0
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2),x,6,256/3465*a*(11*A-5*B)*c^5*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+2/99*a*(11*A-5*B)*c^2*cos(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/f-2/11*a*B*c*cos(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/f+64/1155*a*(11*A-5*B)*c^4*cos(e+f*x)^3/(f*sqrt(c-c*sin(e+f*x)))+8/231*a*(11*A-5*B)*c^3*cos(e+f*x)^3*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2),x,5,64/315*a*(3*A-B)*c^4*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))-2/9*a*B*c*cos(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/f+16/105*a*(3*A-B)*c^3*cos(e+f*x)^3/(f*sqrt(c-c*sin(e+f*x)))+2/21*a*(3*A-B)*c^2*cos(e+f*x)^3*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2),x,4,8/105*a*(7*A-B)*c^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+2/35*a*(7*A-B)*c^2*cos(e+f*x)^3/(f*sqrt(c-c*sin(e+f*x)))-2/7*a*B*c*cos(e+f*x)^3*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2),x,3,2/15*a*(5*A+B)*c^2*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))-2/5*a*B*c*cos(e+f*x)^3/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,5,2*a*(A+B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(f*sqrt(c))-2/3*a*(3*A+5*B)*cos(e+f*x)/(f*sqrt(c-c*sin(e+f*x)))+2/3*a*B*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(c*f)],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,5,a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(3/2))-a*(A+5*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(3/2)*f*sqrt(2))+2*a*B*cos(e+f*x)/(c*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,5,1/2*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(5/2))-1/8*a*(A+9*B)*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(3/2))-1/8*a*(A-7*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(5/2)*f*sqrt(2))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,6,1/3*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(7/2))-1/24*a*(A+13*B)*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(5/2))-1/32*a*(A-3*B)*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(3/2))-1/32*a*(A-3*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(7/2)*f*sqrt(2))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2),x,6,256/15015*a^2*(13*A-3*B)*c^6*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))+64/3003*a^2*(13*A-3*B)*c^5*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(3/2))-2/13*a^2*B*c^2*cos(e+f*x)^5*(c-c*sin(e+f*x))^(3/2)/f+8/429*a^2*(13*A-3*B)*c^4*cos(e+f*x)^5/(f*sqrt(c-c*sin(e+f*x)))+2/143*a^2*(13*A-3*B)*c^3*cos(e+f*x)^5*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2),x,5,64/3465*a^2*(11*A-B)*c^5*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))+16/693*a^2*(11*A-B)*c^4*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(3/2))+2/99*a^2*(11*A-B)*c^3*cos(e+f*x)^5/(f*sqrt(c-c*sin(e+f*x)))-2/11*a^2*B*c^2*cos(e+f*x)^5*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2),x,4,8/315*a^2*(9*A+B)*c^4*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))+2/63*a^2*(9*A+B)*c^3*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(3/2))-2/9*a^2*B*c^2*cos(e+f*x)^5/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2),x,3,2/35*a^2*(7*A+3*B)*c^3*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))-2/7*a^2*B*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(3/2))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,6,-2/5*a^2*B*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))-2/3*a^2*(A+B)*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+4*a^2*(A+B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(f*sqrt(c))-4*a^2*(A+B)*cos(e+f*x)/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,6,1/2*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(7/2))+1/6*a^2*(3*A+7*B)*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))-a^2*(3*A+7*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(c^(3/2)*f)+a^2*(3*A+7*B)*cos(e+f*x)/(c*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,6,1/4*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(9/2))-1/8*a^2*(A+9*B)*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(5/2))+3/4*a^2*(A+9*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(5/2)*f*sqrt(2))-3/8*a^2*(A+9*B)*cos(e+f*x)/(c^2*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,6,1/6*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(11/2))+1/24*a^2*(A-11*B)*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(7/2))-1/16*a^2*(A-11*B)*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(3/2))+1/16*a^2*(A-11*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(7/2)*f*sqrt(2))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(9/2),x,7,1/8*a^2*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(13/2))+1/48*a^2*(3*A-13*B)*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(9/2))-1/64*a^2*(3*A-13*B)*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(5/2))+1/256*a^2*(3*A-13*B)*cos(e+f*x)/(c^3*f*(c-c*sin(e+f*x))^(3/2))+1/256*a^2*(3*A-13*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(9/2)*f*sqrt(2))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2),x,6,256/45045*a^3*(15*A-B)*c^7*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))+64/6435*a^3*(15*A-B)*c^6*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(5/2))+8/715*a^3*(15*A-B)*c^5*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(3/2))+2/195*a^3*(15*A-B)*c^4*cos(e+f*x)^7/(f*sqrt(c-c*sin(e+f*x)))-2/15*a^3*B*c^3*cos(e+f*x)^7*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2),x,5,64/9009*a^3*(13*A+B)*c^6*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))+16/1287*a^3*(13*A+B)*c^5*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(5/2))+2/143*a^3*(13*A+B)*c^4*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(3/2))-2/13*a^3*B*c^3*cos(e+f*x)^7/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2),x,4,8/693*a^3*(11*A+3*B)*c^5*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))+2/99*a^3*(11*A+3*B)*c^4*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(5/2))-2/11*a^3*B*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(3/2))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2),x,3,2/63*a^3*(9*A+5*B)*c^4*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))-2/9*a^3*B*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(5/2))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,7,-2/7*a^3*B*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))-2/5*a^3*(A+B)*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))-4/3*a^3*(A+B)*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+8*a^3*(A+B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(f*sqrt(c))-8*a^3*(A+B)*cos(e+f*x)/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,7,1/2*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(9/2))+1/10*a^3*(5*A+9*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))+1/3*a^3*(5*A+9*B)*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))-2*a^3*(5*A+9*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(c^(3/2)*f)+2*a^3*(5*A+9*B)*cos(e+f*x)/(c*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,7,1/4*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(11/2))-1/8*a^3*(3*A+11*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(7/2))-5/24*a^3*(3*A+11*B)*cos(e+f*x)^3/(c*f*(c-c*sin(e+f*x))^(3/2))+5/2*a^3*(3*A+11*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(5/2)*f*sqrt(2))-5/4*a^3*(3*A+11*B)*cos(e+f*x)/(c^2*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,7,1/6*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(13/2))-1/24*a^3*(A+13*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(9/2))+5/48*a^3*(A+13*B)*cos(e+f*x)^3/(c*f*(c-c*sin(e+f*x))^(5/2))-5/8*a^3*(A+13*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(7/2)*f*sqrt(2))+5/16*a^3*(A+13*B)*cos(e+f*x)/(c^3*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(9/2),x,7,1/8*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(15/2))+1/48*a^3*(A-15*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(11/2))-5/192*a^3*(A-15*B)*cos(e+f*x)^3/(c*f*(c-c*sin(e+f*x))^(7/2))+5/128*a^3*(A-15*B)*cos(e+f*x)/(c^3*f*(c-c*sin(e+f*x))^(3/2))-5/128*a^3*(A-15*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(9/2)*f*sqrt(2))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(11/2),x,8,1/10*a^3*(A+B)*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(17/2))+1/80*a^3*(3*A-17*B)*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(13/2))-1/96*a^3*(3*A-17*B)*cos(e+f*x)^3/(c*f*(c-c*sin(e+f*x))^(9/2))+1/128*a^3*(3*A-17*B)*cos(e+f*x)/(c^3*f*(c-c*sin(e+f*x))^(5/2))-1/512*a^3*(3*A-17*B)*cos(e+f*x)/(c^4*f*(c-c*sin(e+f*x))^(3/2))-1/512*a^3*(3*A-17*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(11/2)*f*sqrt(2))],

# m<0
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x)),x,6,-12/35*(7*A-9*B)*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f)-1/7*(7*A-9*B)*c*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f)-(A-B)*sec(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(a*c*f)-128/35*(7*A-9*B)*c^4*cos(e+f*x)/(a*f*sqrt(c-c*sin(e+f*x)))-32/35*(7*A-9*B)*c^3*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x)),x,5,-1/5*(5*A-7*B)*c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f)-(A-B)*sec(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(a*c*f)-32/15*(5*A-7*B)*c^3*cos(e+f*x)/(a*f*sqrt(c-c*sin(e+f*x)))-8/15*(5*A-7*B)*c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x)),x,4,-(A-B)*sec(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*c*f)-4/3*(3*A-5*B)*c^2*cos(e+f*x)/(a*f*sqrt(c-c*sin(e+f*x)))-1/3*(3*A-5*B)*c*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x)),x,3,-(A-B)*sec(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*c*f)-(A-3*B)*c*cos(e+f*x)/(a*f*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)),x,4,(A+B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a*f*sqrt(2)*sqrt(c))-(A-B)*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*c*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)),x,5,1/4*(3*A-B)*cos(e+f*x)/(a*f*(c-c*sin(e+f*x))^(3/2))+1/4*(3*A-B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a*c^(3/2)*f*sqrt(2))-(A-B)*sec(e+f*x)/(a*c*f*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)),x,6,3/32*(5*A-3*B)*cos(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^(3/2))+1/4*(A+B)*sec(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^(3/2))+3/32*(5*A-3*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a*c^(5/2)*f*sqrt(2))-1/8*(5*A-3*B)*sec(e+f*x)/(a*c^2*f*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(9/2)/(a+a*sin(e+f*x))^2,x,7,-512/105*(7*A-13*B)*c^3*sec(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f)-64/105*(7*A-13*B)*c^2*sec(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a^2*f)-16/105*(7*A-13*B)*c*sec(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(a^2*f)-1/21*(7*A-13*B)*sec(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(a^2*f)-1/3*(A-B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(13/2)/(a^2*c^2*f)+2048/105*(7*A-13*B)*c^4*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^2,x,6,-32/15*(5*A-11*B)*c^2*sec(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f)-4/15*(5*A-11*B)*c*sec(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a^2*f)-1/15*(5*A-11*B)*sec(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(a^2*f)-1/3*(A-B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(11/2)/(a^2*c^2*f)+128/15*(5*A-11*B)*c^3*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^2,x,5,-8/3*(A-3*B)*c*sec(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f)-1/3*(A-3*B)*sec(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a^2*f)-1/3*(A-B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(9/2)/(a^2*c^2*f)+32/3*(A-3*B)*c^2*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^2,x,4,-1/3*(A-7*B)*sec(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f)-1/3*(A-B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(7/2)/(a^2*c^2*f)+4/3*(A-7*B)*c*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^2,x,3,-1/3*(A-B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^2*c^2*f)-1/3*(A+5*B)*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(1/2)),x,5,-1/3*(A-B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^2*c^2*f)+1/2*(A+B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^2*f*sqrt(2)*sqrt(c))-1/2*(A+B)*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*c*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(3/2)),x,6,1/8*(5*A+B)*cos(e+f*x)/(a^2*f*(c-c*sin(e+f*x))^(3/2))+1/8*(5*A+B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^2*c^(3/2)*f*sqrt(2))-1/6*(5*A+B)*sec(e+f*x)/(a^2*c*f*sqrt(c-c*sin(e+f*x)))-1/3*(A-B)*sec(e+f*x)^3*sqrt(c-c*sin(e+f*x))/(a^2*c^2*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(5/2)),x,7,5/64*(7*A-B)*cos(e+f*x)/(a^2*c*f*(c-c*sin(e+f*x))^(3/2))+1/24*(7*A-B)*sec(e+f*x)/(a^2*c*f*(c-c*sin(e+f*x))^(3/2))+5/64*(7*A-B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^2*c^(5/2)*f*sqrt(2))-5/48*(7*A-B)*sec(e+f*x)/(a^2*c^2*f*sqrt(c-c*sin(e+f*x)))-1/3*(A-B)*sec(e+f*x)^3/(a^2*c^2*f*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(9/2)/(a+a*sin(e+f*x))^3,x,7,-2048/15*(A-3*B)*c^3*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^3*f)+512/5*(A-3*B)*c^2*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^3*f)-64/5*(A-3*B)*c*sec(e+f*x)^3*(c-c*sin(e+f*x))^(7/2)/(a^3*f)-16/15*(A-3*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(9/2)/(a^3*f)-1/5*(A-3*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(11/2)/(a^3*c*f)-1/5*(A-B)*sec(e+f*x)^5*(c-c*sin(e+f*x))^(15/2)/(a^3*c^3*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^3,x,6,-128/15*(3*A-13*B)*c^2*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^3*f)+32/5*(3*A-13*B)*c*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^3*f)-4/5*(3*A-13*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(7/2)/(a^3*f)-1/15*(3*A-13*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(9/2)/(a^3*c*f)-1/5*(A-B)*sec(e+f*x)^5*(c-c*sin(e+f*x))^(13/2)/(a^3*c^3*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^3,x,5,-32/15*(A-11*B)*c*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^3*f)+8/5*(A-11*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^3*f)-1/5*(A-11*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(7/2)/(a^3*c*f)-1/5*(A-B)*sec(e+f*x)^5*(c-c*sin(e+f*x))^(11/2)/(a^3*c^3*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^3,x,4,4/15*(A+9*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^3*f)-1/5*(A+9*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^3*c*f)-1/5*(A-B)*sec(e+f*x)^5*(c-c*sin(e+f*x))^(9/2)/(a^3*c^3*f)],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^3,x,3,-1/15*(3*A+7*B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^3*c*f)-1/5*(A-B)*sec(e+f*x)^5*(c-c*sin(e+f*x))^(7/2)/(a^3*c^3*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(1/2)),x,6,-1/6*(A+B)*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^3*c^2*f)-1/5*(A-B)*sec(e+f*x)^5*(c-c*sin(e+f*x))^(5/2)/(a^3*c^3*f)+1/4*(A+B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^3*f*sqrt(2)*sqrt(c))-1/4*(A+B)*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^3*c*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(3/2)),x,7,1/16*(7*A+3*B)*cos(e+f*x)/(a^3*f*(c-c*sin(e+f*x))^(3/2))-1/5*(A-B)*sec(e+f*x)^5*(c-c*sin(e+f*x))^(3/2)/(a^3*c^3*f)+1/16*(7*A+3*B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^3*c^(3/2)*f*sqrt(2))-1/12*(7*A+3*B)*sec(e+f*x)/(a^3*c*f*sqrt(c-c*sin(e+f*x)))-1/30*(7*A+3*B)*sec(e+f*x)^3*sqrt(c-c*sin(e+f*x))/(a^3*c^2*f)],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(5/2)),x,8,7/128*(9*A+B)*cos(e+f*x)/(a^3*c*f*(c-c*sin(e+f*x))^(3/2))+7/240*(9*A+B)*sec(e+f*x)/(a^3*c*f*(c-c*sin(e+f*x))^(3/2))+7/128*(9*A+B)*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^3*c^(5/2)*f*sqrt(2))-7/96*(9*A+B)*sec(e+f*x)/(a^3*c^2*f*sqrt(c-c*sin(e+f*x)))-1/30*(9*A+B)*sec(e+f*x)^3/(a^3*c^2*f*sqrt(c-c*sin(e+f*x)))-1/5*(A-B)*sec(e+f*x)^5*sqrt(c-c*sin(e+f*x))/(a^3*c^3*f)],

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

# m>0
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)),x,3,-1/4*a*(A+B)*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*sqrt(a+a*sin(e+f*x)))+1/5*a*B*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(c*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)),x,3,-1/3*a*(A+B)*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*sqrt(a+a*sin(e+f*x)))+1/4*a*B*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(c*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)),x,3,-1/2*a*(A+B)*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))+1/3*a*B*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(c*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)*sqrt(a+a*sin(e+f*x)),x,3,1/2*a*B*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(c*f*sqrt(a+a*sin(e+f*x)))-a*(A+B)*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,5,-a*(A+B)*cos(e+f*x)*log(1-sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+a*B*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,5,a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+a*B*cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,3,1/2*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-a*B*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,3,1/3*a*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))-1/2*a*B*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2),x,3,-1/6*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/f-1/30*a^2*(3*A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*sqrt(a+a*sin(e+f*x)))-1/15*a*(3*A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2),x,3,-1/5*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)/f-1/30*a^2*(5*A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*sqrt(a+a*sin(e+f*x)))-1/20*a*(5*A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2),x,3,-1/4*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)/f-1/3*a^2*A*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))-1/3*a*A*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2),x,3,1/2*(A-B)*c*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*sqrt(c-c*sin(e+f*x)))+1/3*B*c*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(a*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,5,-1/2*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*sqrt(c-c*sin(e+f*x)))-2*a^2*(A+B)*cos(e+f*x)*log(1-sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-a*(A+B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,5,1/2*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(3/2))+a^2*(A+3*B)*cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+1/2*a*(A+3*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,5,1/4*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(5/2))-a*B*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*(c-c*sin(e+f*x))^(3/2))-a^2*B*cos(e+f*x)*log(1-sin(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,2,1/6*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(7/2))+1/24*(A-5*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(5/2))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(9/2),x,3,1/8*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(9/2))+1/24*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(7/2))+1/96*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^2*f*(c-c*sin(e+f*x))^(5/2))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(11/2),x,3,1/10*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(11/2))-1/120*a^2*(3*A-7*B)*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+1/40*a*(3*A-7*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*(c-c*sin(e+f*x))^(9/2))],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2),x,4,-1/42*a*(7*A-B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/f-1/7*B*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2)/f-1/105*a^3*(7*A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*sqrt(a+a*sin(e+f*x)))-2/105*a^2*(7*A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2),x,4,-1/5*a*A*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)/f-1/6*B*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)/f-2/15*a^3*A*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*sqrt(a+a*sin(e+f*x)))-1/5*a^2*A*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2),x,3,-1/5*B*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/f+1/30*(5*A+B)*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*sqrt(c-c*sin(e+f*x)))+1/20*(5*A+B)*c*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2),x,3,1/3*(A-B)*c*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*sqrt(c-c*sin(e+f*x)))+1/4*B*c*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,6,-1/2*a*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*sqrt(c-c*sin(e+f*x)))-1/3*B*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*sqrt(c-c*sin(e+f*x)))-4*a^3*(A+B)*cos(e+f*x)*log(1-sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2*a^2*(A+B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,6,1/2*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(3/2))+1/2*a*(A+2*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*sqrt(c-c*sin(e+f*x)))+4*a^3*(A+2*B)*cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*a^2*(A+2*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,6,1/4*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(5/2))-1/4*a*(A+5*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(3/2))-a^3*(A+5*B)*cos(e+f*x)*log(1-sin(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-1/2*a^2*(A+5*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,6,1/6*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(7/2))-1/2*a*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(5/2))+a^2*B*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*(c-c*sin(e+f*x))^(3/2))+a^3*B*cos(e+f*x)*log(1-sin(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(9/2),x,2,1/8*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(9/2))+1/48*(A-7*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(7/2))],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(11/2),x,3,1/10*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(11/2))+1/40*(A-4*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(9/2))+1/240*(A-4*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c^2*f*(c-c*sin(e+f*x))^(7/2))],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(13/2),x,4,1/12*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(13/2))+1/40*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(11/2))+1/160*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c^2*f*(c-c*sin(e+f*x))^(9/2))+1/960*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c^3*f*(c-c*sin(e+f*x))^(7/2))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(9/2),x,5,-1/84*a^2*(9*A-B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(9/2)/f-1/72*a*(9*A-B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(9/2)/f-1/9*B*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(9/2)/f-1/315*a^4*(9*A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(f*sqrt(a+a*sin(e+f*x)))-1/126*a^3*(9*A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2),x,5,-1/7*a^2*A*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/f-1/7*a*A*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2)/f-1/8*B*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(7/2)/f-2/35*a^4*A*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*sqrt(a+a*sin(e+f*x)))-4/35*a^3*A*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2),x,4,1/42*(7*A+B)*c*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(3/2)/f-1/7*B*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(5/2)/f+1/105*(7*A+B)*c^3*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*sqrt(c-c*sin(e+f*x)))+2/105*(7*A+B)*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2),x,3,-1/6*B*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(3/2)/f+1/30*(3*A+B)*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*sqrt(c-c*sin(e+f*x)))+1/15*(3*A+B)*c*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2),x,3,1/4*(A-B)*c*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*sqrt(c-c*sin(e+f*x)))+1/5*B*c*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,7,-a^2*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*sqrt(c-c*sin(e+f*x)))-1/3*a*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*sqrt(c-c*sin(e+f*x)))-1/4*B*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*sqrt(c-c*sin(e+f*x)))-8*a^4*(A+B)*cos(e+f*x)*log(1-sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-4*a^3*(A+B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,7,1/2*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(3/2))+1/2*a^2*(3*A+5*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*sqrt(c-c*sin(e+f*x)))+1/6*a*(3*A+5*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*sqrt(c-c*sin(e+f*x)))+4*a^4*(3*A+5*B)*cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*a^3*(3*A+5*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,7,1/4*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(5/2))-1/2*a*(A+3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(3/2))-3/4*a^2*(A+3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^2*f*sqrt(c-c*sin(e+f*x)))-6*a^4*(A+3*B)*cos(e+f*x)*log(1-sin(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-3*a^3*(A+3*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,7,1/6*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(7/2))-1/12*a*(A+7*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(5/2))+1/4*a^2*(A+7*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^2*f*(c-c*sin(e+f*x))^(3/2))+a^4*(A+7*B)*cos(e+f*x)*log(1-sin(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+1/2*a^3*(A+7*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^3*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(9/2),x,7,1/8*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(9/2))-1/3*a*B*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(7/2))+1/2*a^2*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^2*f*(c-c*sin(e+f*x))^(5/2))-a^3*B*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^3*f*(c-c*sin(e+f*x))^(3/2))-a^4*B*cos(e+f*x)*log(1-sin(e+f*x))/(c^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(11/2),x,2,1/10*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(11/2))+1/80*(A-9*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(9/2))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(13/2),x,3,1/12*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(13/2))+1/60*(A-5*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(11/2))+1/480*(A-5*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c^2*f*(c-c*sin(e+f*x))^(9/2))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(15/2),x,4,1/14*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(15/2))+1/168*(3*A-11*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(13/2))+1/840*(3*A-11*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c^2*f*(c-c*sin(e+f*x))^(11/2))+1/6720*(3*A-11*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c^3*f*(c-c*sin(e+f*x))^(9/2))],
[(a+a*sin(e+f*x))^(7/2)*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(17/2),x,5,1/16*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(17/2))+1/56*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(15/2))+1/224*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c^2*f*(c-c*sin(e+f*x))^(13/2))+1/1120*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c^3*f*(c-c*sin(e+f*x))^(11/2))+1/8960*(A-3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c^4*f*(c-c*sin(e+f*x))^(9/2))],

# m<0
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)/sqrt(a+a*sin(e+f*x)),x,6,1/2*(A-B)*c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))-1/3*B*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*sqrt(a+a*sin(e+f*x)))+4*(A-B)*c^3*cos(e+f*x)*log(1+sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*(A-B)*c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)/sqrt(a+a*sin(e+f*x)),x,5,-1/2*B*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))+2*(A-B)*c^2*cos(e+f*x)*log(1+sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+(A-B)*c*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)/sqrt(a+a*sin(e+f*x)),x,5,(A-B)*c*cos(e+f*x)*log(1+sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-B*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((c-c*sin(e+f*x))^(1/2)*sqrt(a+a*sin(e+f*x))),x,7,-1/2*(A+B)*cos(e+f*x)*log(1-sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+1/2*(A-B)*cos(e+f*x)*log(1+sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))),x,3,1/2*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/2*(A-B)*arctanh(sin(e+f*x))*cos(e+f*x)/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))),x,4,1/4*(A+B)*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+1/4*(A-B)*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/4*(A-B)*arctanh(sin(e+f*x))*cos(e+f*x)/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^(3/2),x,7,-1/2*(A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(3*A-5*B)*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*sqrt(a+a*sin(e+f*x)))-1/6*(3*A-5*B)*c*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f*sqrt(a+a*sin(e+f*x)))-4*(3*A-5*B)*c^4*cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2*(3*A-5*B)*c^3*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(3/2),x,6,-1/2*(A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(A-2*B)*c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*sqrt(a+a*sin(e+f*x)))-4*(A-2*B)*c^3*cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2*(A-2*B)*c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(3/2),x,5,-1/2*(A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^(3/2))-(A-3*B)*c^2*cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-1/2*(A-3*B)*c*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^(3/2),x,5,-(A-B)*c*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))+B*c*cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(1/2)),x,3,-1/2*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))+1/2*(A+B)*arctanh(sin(e+f*x))*cos(e+f*x)/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)),x,4,-1/2*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2))+1/2*A*cos(e+f*x)/(a*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/2*A*arctanh(sin(e+f*x))*cos(e+f*x)/(a*c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)),x,5,-1/2*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2))+1/8*(3*A-B)*cos(e+f*x)/(a*f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+1/8*(3*A-B)*cos(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/8*(3*A-B)*arctanh(sin(e+f*x))*cos(e+f*x)/(a*c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(9/2)/(a+a*sin(e+f*x))^(5/2),x,8,1/4*(3*A-7*B)*c*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/4*(A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(f*(a+a*sin(e+f*x))^(5/2))+(3*A-7*B)*c^3*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))+1/3*(3*A-7*B)*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))+8*(3*A-7*B)*c^5*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+4*(3*A-7*B)*c^4*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^(5/2),x,7,1/2*(A-3*B)*c*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/4*(A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*(a+a*sin(e+f*x))^(5/2))+3/4*(A-3*B)*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))+6*(A-3*B)*c^4*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+3*(A-3*B)*c^3*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(5/2),x,6,1/4*(A-5*B)*c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/4*(A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*(a+a*sin(e+f*x))^(5/2))+(A-5*B)*c^3*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+1/2*(A-5*B)*c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(5/2),x,5,-1/4*(A-B)*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^(5/2))-B*c^2*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-B*c*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*(a+a*sin(e+f*x))^(3/2))],
[(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^(5/2),x,3,-1/2*(A-B)*c*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)))-B*c*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(1/2)),x,4,-1/4*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)))-1/4*(A+B)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))+1/4*(A+B)*arctanh(sin(e+f*x))*cos(e+f*x)/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)),x,5,-1/4*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2))-1/8*(3*A+B)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2))+1/8*(3*A+B)*cos(e+f*x)/(a^2*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/8*(3*A+B)*arctanh(sin(e+f*x))*cos(e+f*x)/(a^2*c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)),x,6,-1/4*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2))-1/2*A*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2))+3/8*A*cos(e+f*x)/(a^2*f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+3/8*A*cos(e+f*x)/(a^2*c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+3/8*A*arctanh(sin(e+f*x))*cos(e+f*x)/(a^2*c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],

# Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x]) when m and/or n symbolic
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^n,x,5,2^(1/2+n)*c*(B*(m-n)+A*(1+m+n))*cos(e+f*x)*hypergeom([1/2*(1+2*m),1/2*(1-2*n)],[1/2*(3+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n)/(f*(1+2*m)*(1+m+n))-B*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(f*(1+m+n))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^3,x,5,1/7*2^(1/2+m)*a^4*c^3*(B*(3-m)-A*(4+m))*cos(e+f*x)^7*hypergeom([7/2,1/2-m],[9/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^(-4+m)/(f*(4+m))-a^3*B*c^3*cos(e+f*x)^7*(a+a*sin(e+f*x))^(-3+m)/(f*(4+m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^2,x,5,1/5*2^(1/2+m)*a^3*c^2*(B*(2-m)-A*(3+m))*cos(e+f*x)^5*hypergeom([5/2,1/2-m],[7/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^(-3+m)/(f*(3+m))-a^2*B*c^2*cos(e+f*x)^5*(a+a*sin(e+f*x))^(-2+m)/(f*(3+m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x)),x,5,1/3*2^(1/2+m)*a^2*c*(B*(1-m)-A*(2+m))*cos(e+f*x)^3*hypergeom([3/2,1/2-m],[5/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^(-2+m)/(f*(2+m))-a*B*c*cos(e+f*x)^3*(a+a*sin(e+f*x))^(-1+m)/(f*(2+m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,-B*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+m))-2^(1/2+m)*(A+A*m+B*m)*cos(e+f*x)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(f*(1+m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c-c*sin(e+f*x)),x,5,2^(1/2+m)*(B+A*m+B*m)*hypergeom([-1/2,1/2-m],[1/2],1/2*(1-sin(e+f*x)))*sec(e+f*x)*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^m/(c*f*m)-B*sec(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*c*f*m)],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^2,x,5,1/3*2^(1/2+m)*(A*(1-m)-B*(2+m))*hypergeom([-3/2,1/2-m],[-1/2],1/2*(1-sin(e+f*x)))*sec(e+f*x)^3*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^(1+m)/(a*c^2*f*(1-m))+B*sec(e+f*x)^3*(a+a*sin(e+f*x))^(2+m)/(a^2*c^2*f*(1-m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^3,x,5,1/5*2^(1/2+m)*(A*(2-m)-B*(3+m))*hypergeom([-5/2,1/2-m],[-3/2],1/2*(1-sin(e+f*x)))*sec(e+f*x)^5*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^(2+m)/(a^2*c^3*f*(2-m))+B*sec(e+f*x)^5*(a+a*sin(e+f*x))^(3+m)/(a^3*c^3*f*(2-m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/sqrt(c-c*sin(e+f*x)),x,4,-2*B*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+2*m)*sqrt(c-c*sin(e+f*x)))+(A+B)*cos(e+f*x)*hypergeom([1,1/2+m],[3/2+m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^m/(f*(1+2*m)*sqrt(c-c*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+c*sin(e+f*x))^m/sqrt(a-a*sin(e+f*x)),x,4,-2*B*cos(e+f*x)*(c+c*sin(e+f*x))^m/(f*(1+2*m)*sqrt(a-a*sin(e+f*x)))+(A+B)*cos(e+f*x)*hypergeom([1,1/2+m],[3/2+m],1/2*(1+sin(e+f*x)))*(c+c*sin(e+f*x))^m/(f*(1+2*m)*sqrt(a-a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2),x,4,-2*c*(B*(5-2*m)-A*(7+2*m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(3/2)/(f*(35+24*m+4*m^2))-2*B*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(5/2)/(f*(7+2*m))-64*c^3*(B*(5-2*m)-A*(7+2*m))*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(5+2*m)*(7+2*m)*(3+8*m+4*m^2)*sqrt(c-c*sin(e+f*x)))-16*c^2*(B*(5-2*m)-A*(7+2*m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(c-c*sin(e+f*x))/(f*(7+2*m)*(15+16*m+4*m^2))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2),x,3,4*(A-B)*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+2*m)*sqrt(c-c*sin(e+f*x)))-2*(A-3*B)*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(3+2*m)*sqrt(c-c*sin(e+f*x)))-2*B*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(2+m)/(a^2*f*(5+2*m)*sqrt(c-c*sin(e+f*x))),-2*B*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(3/2)/(f*(5+2*m))-8*c^2*(B*(3-2*m)-A*(5+2*m))*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(5+2*m)*(3+8*m+4*m^2)*sqrt(c-c*sin(e+f*x)))-2*c*(B*(3-2*m)-A*(5+2*m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(c-c*sin(e+f*x))/(f*(15+16*m+4*m^2))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2),x,3,2*(A-B)*c*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+2*m)*sqrt(c-c*sin(e+f*x)))+2*B*c*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(3+2*m)*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,4,-2*B*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+2*m)*sqrt(c-c*sin(e+f*x)))+(A+B)*cos(e+f*x)*hypergeom([1,1/2+m],[3/2+m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^m/(f*(1+2*m)*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,4,1/2*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(c-c*sin(e+f*x))^(3/2))+1/4*(A*(1-2*m)-B*(3+2*m))*cos(e+f*x)*hypergeom([1,1/2+m],[3/2+m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^m/(c*f*(1+2*m)*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,4,1/4*(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(c-c*sin(e+f*x))^(5/2))+1/16*(A*(3-2*m)-B*(5+2*m))*cos(e+f*x)*hypergeom([2,1/2+m],[3/2+m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^m/(c^2*f*(1+2*m)*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(-4-m),x,4,(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-4-m)/(f*(7+2*m))+(3*A-2*B*(2+m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-3-m)/(c*f*(35+24*m+4*m^2))+2*(3*A-2*B*(2+m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m)/(c^2*f*(105+142*m+60*m^2+8*m^3))+2*(3*A-2*B*(2+m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(c^3*f*(105+352*m+344*m^2+128*m^3+16*m^4))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(-3-m),x,3,(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-3-m)/(f*(5+2*m))+(2*A-B*(3+2*m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m)/(c*f*(15+16*m+4*m^2))+(2*A-B*(3+2*m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(c^2*f*(15+46*m+36*m^2+8*m^3))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(-2-m),x,2,(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m)/(f*(3+2*m))+(A-2*B*(1+m))*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(c*f*(3+8*m+4*m^2))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(-1-m),x,5,(A+B)*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*(1+2*m))-2^(1/2-m)*B*cos(e+f*x)*hypergeom([1/2*(1+2*m),1/2*(1+2*m)],[1/2*(3+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*(1+2*m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^m,x,5,2^(1/2-m)*c*(A+2*B*m)*cos(e+f*x)*hypergeom([1/2*(1+2*m),1/2*(1+2*m)],[1/2*(3+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*(1+2*m))-B*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(c-c*sin(e+f*x))^m)],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1-m),x,5,2^(1/2-m)*c^2*(2*A-B*(1-2*m))*cos(e+f*x)*hypergeom([1/2*(-1+2*m),1/2*(1+2*m)],[1/2*(3+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*(1+2*m))-1/2*B*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(1-m)/f],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(2-m),x,5,1/3*2^(5/2-m)*c^3*(3*A-2*B*(1-m))*cos(e+f*x)*hypergeom([1/2*(-3+2*m),1/2*(1+2*m)],[1/2*(3+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*(1+2*m))-1/3*B*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(2-m)/f],

#  Degenerate special cases 
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^n*(B*(3-n)-B*(4+n)*sin(e+f*x)),x,2,a^3*B*c^3*cos(e+f*x)^7*(c-c*sin(e+f*x))^(-3+n)/f],
[(a-a*sin(e+f*x))^3*(c+c*sin(e+f*x))^n*(B*(3-n)+B*(4+n)*sin(e+f*x)),x,2,-a^3*B*c^3*cos(e+f*x)^7*(c+c*sin(e+f*x))^(-3+n)/f],
[(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^3*(B*(-3+m)-B*(4+m)*sin(e+f*x)),x,2,a^3*B*c^3*cos(e+f*x)^7*(a+a*sin(e+f*x))^(-3+m)/f],
[(a-a*sin(e+f*x))^m*(c+c*sin(e+f*x))^3*(B*(-3+m)+B*(4+m)*sin(e+f*x)),x,2,-a^3*B*c^3*cos(e+f*x)^7*(a-a*sin(e+f*x))^(-3+m)/f],
[(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n*(B*(m-n)-B*(1+m+n)*sin(e+f*x)),x,1,B*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/f],
[(a-a*sin(e+f*x))^m*(c+c*sin(e+f*x))^n*(B*(m-n)+B*(1+m+n)*sin(e+f*x)),x,1,-B*cos(e+f*x)*(a-a*sin(e+f*x))^m*(c+c*sin(e+f*x))^n/f],

# Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x])

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

# m>0
[sin(c+d*x)^3*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,13,1/8*a^3*A*x-2/3*a^3*A*cos(c+d*x)^3/d+3/5*a^3*A*cos(c+d*x)^5/d-1/7*a^3*A*cos(c+d*x)^7/d-1/8*a^3*A*cos(c+d*x)*sin(c+d*x)/d-1/12*a^3*A*cos(c+d*x)*sin(c+d*x)^3/d+1/3*a^3*A*cos(c+d*x)*sin(c+d*x)^5/d],
[sin(c+d*x)^2*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,12,3/16*a^3*A*x-2/3*a^3*A*cos(c+d*x)^3/d+2/5*a^3*A*cos(c+d*x)^5/d-3/16*a^3*A*cos(c+d*x)*sin(c+d*x)/d+5/24*a^3*A*cos(c+d*x)*sin(c+d*x)^3/d+1/6*a^3*A*cos(c+d*x)*sin(c+d*x)^5/d],
[sin(c+d*x)*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,10,1/4*a^3*A*x-2/3*a^3*A*cos(c+d*x)^3/d+1/5*a^3*A*cos(c+d*x)^5/d-1/4*a^3*A*cos(c+d*x)*sin(c+d*x)/d+1/2*a^3*A*cos(c+d*x)*sin(c+d*x)^3/d],
[(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,5,5/8*a^3*A*x-5/12*a^3*A*cos(c+d*x)^3/d+5/8*a^3*A*cos(c+d*x)*sin(c+d*x)/d-1/4*A*cos(c+d*x)^3*(a^3+a^3*sin(c+d*x))/d],
[csc(c+d*x)*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,7,a^3*A*x-a^3*A*arctanh(cos(c+d*x))/d+a^3*A*cos(c+d*x)/d-1/3*a^3*A*cos(c+d*x)^3/d+a^3*A*cos(c+d*x)*sin(c+d*x)/d],
[csc(c+d*x)^2*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,9,-1/2*a^3*A*x-2*a^3*A*arctanh(cos(c+d*x))/d+2*a^3*A*cos(c+d*x)/d-a^3*A*cot(c+d*x)/d+1/2*a^3*A*cos(c+d*x)*sin(c+d*x)/d],
[csc(c+d*x)^3*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,7,-2*a^3*A*x-1/2*a^3*A*arctanh(cos(c+d*x))/d+a^3*A*cos(c+d*x)/d-2*a^3*A*cot(c+d*x)/d-1/2*a^3*A*cot(c+d*x)*csc(c+d*x)/d],
[csc(c+d*x)^4*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,7,-a^3*A*x+a^3*A*arctanh(cos(c+d*x))/d-a^3*A*cot(c+d*x)/d-1/3*a^3*A*cot(c+d*x)^3/d-a^3*A*cot(c+d*x)*csc(c+d*x)/d],
[csc(c+d*x)^5*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,10,5/8*a^3*A*arctanh(cos(c+d*x))/d-2/3*a^3*A*cot(c+d*x)^3/d-3/8*a^3*A*cot(c+d*x)*csc(c+d*x)/d-1/4*a^3*A*cot(c+d*x)*csc(c+d*x)^3/d],
[csc(c+d*x)^6*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,12,1/4*a^3*A*arctanh(cos(c+d*x))/d-2/3*a^3*A*cot(c+d*x)^3/d-1/5*a^3*A*cot(c+d*x)^5/d+1/4*a^3*A*cot(c+d*x)*csc(c+d*x)/d-1/2*a^3*A*cot(c+d*x)*csc(c+d*x)^3/d],
[csc(c+d*x)^7*(a+a*sin(c+d*x))^3*(A-A*sin(c+d*x)),x,12,3/16*a^3*A*arctanh(cos(c+d*x))/d-2/3*a^3*A*cot(c+d*x)^3/d-2/5*a^3*A*cot(c+d*x)^5/d+3/16*a^3*A*cot(c+d*x)*csc(c+d*x)/d-5/24*a^3*A*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^3*A*cot(c+d*x)*csc(c+d*x)^5/d],

# m<0
[sin(c+d*x)^4*(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,11,-19/2*A*x/a^3-4*A*cos(c+d*x)/(a^3*d)+1/2*A*cos(c+d*x)*sin(c+d*x)/(a^3*d)-2/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^3)+41/15*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)-199/15*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[sin(c+d*x)^3*(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,9,4*A*x/a^3+A*cos(c+d*x)/(a^3*d)+2/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^3)-31/15*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)+104/15*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[sin(c+d*x)^2*(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,8,-A*x/a^3-2/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^3)+7/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)-13/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[sin(c+d*x)*(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,8,2/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^3)-11/15*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)+4/15*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,3,-1/5*a*A*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^4)-1/15*A*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^3)],
[csc(c+d*x)*(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,9,-A*arctanh(cos(c+d*x))/(a^3*d)+2/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^3)+3/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)+8/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[csc(c+d*x)^2*(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,16,4*A*arctanh(cos(c+d*x))/(a^3*d)-94/15*A*cot(c+d*x)/(a^3*d)+2/5*A*cot(c+d*x)/(a^3*d*(1+sin(c+d*x))^3)+13/15*A*cot(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)+4*A*cot(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[csc(c+d*x)^3*(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,13,-19/2*A*arctanh(cos(c+d*x))/(a^3*d)+4*A*cot(c+d*x)/(a^3*d)-1/2*A*cot(c+d*x)*csc(c+d*x)/(a^3*d)+2/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^3)+29/15*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)+164/15*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[csc(c+d*x)^4*(A-A*sin(c+d*x))/(a+a*sin(c+d*x))^3,x,15,18*A*arctanh(cos(c+d*x))/(a^3*d)-10*A*cot(c+d*x)/(a^3*d)-1/3*A*cot(c+d*x)^3/(a^3*d)+2*A*cot(c+d*x)*csc(c+d*x)/(a^3*d)-2/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^3)-13/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)-93/5*A*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],

# Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x])

# m>0
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3,x,5,1/8*a*(B*(4*c^3+12*c^2*d+9*c*d^2+3*d^3)+A*(8*c^3+12*c^2*d+12*c*d^2+3*d^3))*x-1/30*a*(5*A*d*(3*c^3+16*c^2*d+12*c*d^2+4*d^3)-B*(3*c^4-15*c^3*d-52*c^2*d^2-60*c*d^3-16*d^4))*cos(e+f*x)/(d*f)-1/120*a*(5*A*d*(6*c^2+20*c*d+9*d^2)-B*(6*c^3-30*c^2*d-71*c*d^2-45*d^3))*cos(e+f*x)*sin(e+f*x)/f-1/60*a*(4*(5*A+4*B)*d^2-3*c*(B*c-5*(A+B)*d))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d*f)+1/20*a*(B*c-5*(A+B)*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d*f)-1/5*a*B*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d*f)],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2,x,4,1/8*a*(4*A*(2*c^2+2*c*d+d^2)+B*(4*c^2+8*c*d+3*d^2))*x-1/6*a*(4*A*d*(c^2+3*c*d+d^2)-B*(c^3-4*c^2*d-8*c*d^2-4*d^3))*cos(e+f*x)/(d*f)-1/24*a*(3*(4*A+3*B)*d^2-2*c*(B*c-4*(A+B)*d))*cos(e+f*x)*sin(e+f*x)/f+1/12*a*(B*c-4*(A+B)*d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d*f)-1/4*a*B*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d*f)],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c+d*sin(e+f*x)),x,3,1/2*a*(B*(c+d)+A*(2*c+d))*x-1/3*a*(3*A*(c+d)+B*(3*c+d))*cos(e+f*x)/f-1/6*a*(3*B*c+3*A*d-B*d)*cos(e+f*x)*sin(e+f*x)/f-1/3*B*d*cos(e+f*x)*(a+a*sin(e+f*x))^2/(a*f)],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x)),x,1,1/2*a*(2*A+B)*x-a*(A+B)*cos(e+f*x)/f-1/2*a*B*cos(e+f*x)*sin(e+f*x)/f],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c+d*sin(e+f*x)),x,6,-a*(B*c-(A+B)*d)*x/d^2-a*B*cos(e+f*x)/(d*f)+2*a*(c-d)*(B*c-A*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^2*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,6,a*B*x/d^2+2*a*((A+B)*(c-d)*d^2-B*c*(c^2-d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^2*(c^2-d^2)^(3/2)*f)+a*(B*c-A*d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,7,a*(2*A*c+B*c-A*d-2*B*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c+d)*(c^2-d^2)^(3/2)*f)+1/2*a*(B*c-A*d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^2)-1/2*a*(A*(c-2*d)*d+B*(c^2+2*c*d-2*d^2))*cos(e+f*x)/((c-d)*d*(c+d)^2*f*(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3,x,6,1/16*a^2*(6*A*(4*c^3+8*c^2*d+7*c*d^2+2*d^3)+B*(16*c^3+42*c^2*d+36*c*d^2+11*d^3))*x+1/60*a^2*(6*A*d*(c^4-10*c^3*d-44*c^2*d^2-40*c*d^3-12*d^4)-B*(2*c^5-12*c^4*d+47*c^3*d^2+208*c^2*d^3+216*c*d^4+64*d^5))*cos(e+f*x)/(d^2*f)+1/240*a^2*(6*A*d*(2*c^3-20*c^2*d-57*c*d^2-30*d^3)-B*(4*c^4-24*c^3*d+96*c^2*d^2+284*c*d^3+165*d^4))*cos(e+f*x)*sin(e+f*x)/(d*f)+1/120*a^2*(6*A*d*(c^2-10*c*d-12*d^2)-B*(2*c^3-12*c^2*d+51*c*d^2+64*d^3))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d^2*f)+1/120*a^2*(6*A*(c-10*d)*d-B*(2*c^2-12*c*d+55*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^2*f)+1/30*a^2*(2*B*c-6*A*d-7*B*d)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d^2*f)-1/6*B*cos(e+f*x)*(a^2+a^2*sin(e+f*x))*(c+d*sin(e+f*x))^4/(d*f)],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2,x,5,1/8*a^2*(12*A*c^2+8*B*c^2+16*A*c*d+14*B*c*d+7*A*d^2+6*B*d^2)*x+1/30*a^2*(5*A*d*(c^3-8*c^2*d-20*c*d^2-8*d^3)-2*B*(c^4-5*c^3*d+16*c^2*d^2+40*c*d^3+18*d^4))*cos(e+f*x)/(d^2*f)+1/120*a^2*(5*A*d*(2*c^2-16*c*d-21*d^2)-B*(4*c^3-20*c^2*d+66*c*d^2+90*d^3))*cos(e+f*x)*sin(e+f*x)/(d*f)+1/60*a^2*(5*A*(c-8*d)*d-2*B*(c^2-5*c*d+18*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d^2*f)+1/20*a^2*(2*B*(c-3*d)-5*A*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^2*f)-1/5*B*cos(e+f*x)*(a^2+a^2*sin(e+f*x))*(c+d*sin(e+f*x))^3/(d*f)],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c+d*sin(e+f*x)),x,4,1/8*a^2*(12*A*c+8*B*c+8*A*d+7*B*d)*x-1/6*a^2*(12*A*c+8*B*c+8*A*d+7*B*d)*cos(e+f*x)/f-1/24*a^2*(12*A*c+8*B*c+8*A*d+7*B*d)*cos(e+f*x)*sin(e+f*x)/f-1/12*(4*B*c+4*A*d-B*d)*cos(e+f*x)*(a+a*sin(e+f*x))^2/f-1/4*B*d*cos(e+f*x)*(a+a*sin(e+f*x))^3/(a*f)],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x)),x,2,1/2*a^2*(3*A+2*B)*x-2/3*a^2*(3*A+2*B)*cos(e+f*x)/f-1/6*a^2*(3*A+2*B)*cos(e+f*x)*sin(e+f*x)/f-1/3*B*cos(e+f*x)*(a+a*sin(e+f*x))^2/f],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c+d*sin(e+f*x)),x,7,-1/2*a^2*(2*A*(c-2*d)*d-B*(2*c^2-4*c*d+3*d^2))*x/d^3+1/2*a^2*(2*B*c-2*A*d-3*B*d)*cos(e+f*x)/(d^2*f)-1/2*B*cos(e+f*x)*(a^2+a^2*sin(e+f*x))/(d*f)-2*a^2*(c-d)^2*(B*c-A*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^3*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,7,-a^2*(2*B*c-A*d-2*B*d)*x/d^3+a^2*(A*d-B*(2*c+d))*cos(e+f*x)/(d^2*(c+d)*f)+(B*c-A*d)*cos(e+f*x)*(a^2+a^2*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x)))-2*a^2*(c-d)*(A*d*(c+2*d)-B*(2*c^2+2*c*d-d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^3*(c+d)*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,7,a^2*B*x/d^3+1/2*(B*c-A*d)*cos(e+f*x)*(a^2+a^2*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^2)-1/2*a^2*(3*A*d^2-B*(2*c^2+3*c*d-2*d^2))*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x)))+a^2*(3*A*d^3-B*(2*c^3+4*c^2*d+c*d^2-4*d^3))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^3*(c+d)^2*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3,x,7,1/16*a^3*(3*B*(10*c^3+26*c^2*d+23*c*d^2+7*d^3)+A*(40*c^3+90*c^2*d+78*c*d^2+23*d^3))*x-1/420*a^3*(7*A*d*(2*c^5-18*c^4*d+107*c^3*d^2+472*c^2*d^3+456*c*d^4+136*d^5)-3*B*(2*c^6-14*c^5*d+51*c^4*d^2-189*c^3*d^3-920*c^2*d^4-952*c*d^5-288*d^6))*cos(e+f*x)/(d^3*f)-1/1680*a^3*(7*A*d*(4*c^4-36*c^3*d+216*c^2*d^2+626*c*d^3+345*d^4)-3*B*(4*c^5-28*c^4*d+104*c^3*d^2-392*c^2*d^3-1263*c*d^4-735*d^5))*cos(e+f*x)*sin(e+f*x)/(d^2*f)-1/840*a^3*(7*A*d*(2*c^3-18*c^2*d+111*c*d^2+136*d^3)-B*(6*c^4-42*c^3*d+165*c^2*d^2-651*c*d^3-864*d^4))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d^3*f)-1/840*a^3*(7*A*d*(2*c^2-18*c*d+115*d^2)-B*(6*c^3-42*c^2*d+177*c*d^2-735*d^3))*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^3*f)-1/210*a^3*(6*B*c^2-14*A*c*d-27*B*c*d+91*A*d^2+87*B*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d^3*f)-1/7*a*B*cos(e+f*x)*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^4/(d*f)+1/42*(3*B*(c-3*d)-7*A*d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x))^4/(d^2*f)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2,x,6,1/16*a^3*(B*(30*c^2+52*c*d+23*d^2)+A*(40*c^2+60*c*d+26*d^2))*x-1/60*a^3*(2*A*d*(2*c^4-15*c^3*d+72*c^2*d^2+180*c*d^3+76*d^4)-B*(2*c^5-12*c^4*d+37*c^3*d^2-112*c^2*d^3-304*c*d^4-136*d^5))*cos(e+f*x)/(d^3*f)-1/240*a^3*(2*A*d*(4*c^3-30*c^2*d+146*c*d^2+195*d^3)-B*(4*c^4-24*c^3*d+76*c^2*d^2-236*c*d^3-345*d^4))*cos(e+f*x)*sin(e+f*x)/(d^2*f)-1/120*a^3*(2*A*d*(2*c^2-15*c*d+76*d^2)-B*(2*c^3-12*c^2*d+41*c*d^2-136*d^3))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d^3*f)+1/40*a^3*(2*A*(2*c-11*d)*d-B*(2*c^2-8*c*d+21*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^3*f)-1/6*a*B*cos(e+f*x)*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^3/(d*f)+1/30*(3*B*c-6*A*d-8*B*d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x))^3/(d^2*f)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))*(c+d*sin(e+f*x)),x,10,1/8*a^3*(20*A*c+15*B*c+15*A*d+13*B*d)*x-1/5*a^3*(20*A*c+15*B*c+15*A*d+13*B*d)*cos(e+f*x)/f+1/60*a^3*(20*A*c+15*B*c+15*A*d+13*B*d)*cos(e+f*x)^3/f-3/40*a^3*(20*A*c+15*B*c+15*A*d+13*B*d)*cos(e+f*x)*sin(e+f*x)/f-1/20*(5*B*c+5*A*d-B*d)*cos(e+f*x)*(a+a*sin(e+f*x))^3/f-1/5*B*d*cos(e+f*x)*(a+a*sin(e+f*x))^4/(a*f)],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x)),x,8,5/8*a^3*(4*A+3*B)*x-5/6*a^3*(4*A+3*B)*cos(e+f*x)/f-5/24*a^3*(4*A+3*B)*cos(e+f*x)*sin(e+f*x)/f-1/12*a*(4*A+3*B)*cos(e+f*x)*(a+a*sin(e+f*x))^2/f-1/4*B*cos(e+f*x)*(a+a*sin(e+f*x))^3/f,5/8*a^3*(4*A+3*B)*x-a^3*(4*A+3*B)*cos(e+f*x)/f+1/12*a^3*(4*A+3*B)*cos(e+f*x)^3/f-3/8*a^3*(4*A+3*B)*cos(e+f*x)*sin(e+f*x)/f-1/4*B*cos(e+f*x)*(a+a*sin(e+f*x))^3/f],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c+d*sin(e+f*x)),x,8,1/2*a^3*(A*d*(2*c^2-6*c*d+7*d^2)-B*(2*c^3-6*c^2*d+7*c*d^2-5*d^3))*x/d^4+1/2*a^3*(A*(2*c-5*d)*d-B*(2*c^2-5*c*d+5*d^2))*cos(e+f*x)/(d^3*f)-1/3*a*B*cos(e+f*x)*(a+a*sin(e+f*x))^2/(d*f)+1/6*(3*B*c-3*A*d-5*B*d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d^2*f)+2*a^3*(c-d)^3*(B*c-A*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^4*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,8,-1/2*a^3*(2*A*(2*c-3*d)*d-B*(6*c^2-12*c*d+7*d^2))*x/d^4-1/2*a^3*(4*A*c*d-B*(6*c^2-3*c*d-5*d^2))*cos(e+f*x)/(d^3*(c+d)*f)+1/2*(2*A*d-B*(3*c+d))*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d^2*(c+d)*f)+a*(B*c-A*d)*cos(e+f*x)*(a+a*sin(e+f*x))^2/(d*(c+d)*f*(c+d*sin(e+f*x)))+2*a^3*(c-d)^2*(A*d*(2*c+3*d)-B*(3*c^2+3*c*d-d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^4*(c+d)*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^3*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,8,-a^3*(3*B*c-A*d-3*B*d)*x/d^4-1/2*a^3*(3*B*c*(2*c+3*d)-A*d*(2*c+5*d))*cos(e+f*x)/(d^3*(c+d)^2*f)+1/2*a*(B*c-A*d)*cos(e+f*x)*(a+a*sin(e+f*x))^2/(d*(c+d)*f*(c+d*sin(e+f*x))^2)-1/2*(A*d*(c+4*d)-B*(3*c^2+4*c*d-2*d^2))*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d^2*(c+d)^2*f*(c+d*sin(e+f*x)))-a^3*(c-d)*(A*d*(2*c^2+6*c*d+7*d^2)-3*B*(2*c^3+4*c^2*d+c*d^2-2*d^3))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^4*(c+d)^2*f*sqrt(c^2-d^2))],

# m<0
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x)),x,3,1/2*(3*A*d*(2*c^2-2*c*d+d^2)+B*(2*c^3-6*c^2*d+9*c*d^2-3*d^3))*x/a+2/3*d*(3*A*(c^2-3*c*d+d^2)-B*(7*c^2-9*c*d+4*d^2))*cos(e+f*x)/(a*f)+1/6*d^2*(6*A*c-11*B*c-9*A*d+9*B*d)*cos(e+f*x)*sin(e+f*x)/(a*f)+1/3*(3*A-4*B)*d*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a*f)-(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x)),x,2,1/2*(2*A*(2*c-d)*d+B*(2*c^2-4*c*d+3*d^2))*x/a+2*(A*(c-d)-B*(2*c-d))*d*cos(e+f*x)/(a*f)+1/2*(2*A-3*B)*d^2*cos(e+f*x)*sin(e+f*x)/(a*f)-(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))/(a+a*sin(e+f*x)),x,4,(B*(c-d)+A*d)*x/a-B*d*cos(e+f*x)/(a*f)-(A-B)*(c-d)*cos(e+f*x)/(a*f*(1+sin(e+f*x)))],
[(A+B*sin(e+f*x))/(a+a*sin(e+f*x)),x,2,B*x/a-(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))),x,5,-(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x)))+2*(B*c-A*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a*(c-d)*f*sqrt(c^2-d^2))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))^2),x,6,-2*(A*d*(2*c+d)-B*(c^2+c*d+d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a*(c-d)*(c^2-d^2)^(3/2)*f)+d*(B*(2*c+d)-A*(c+2*d))*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sin(e+f*x)))-(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))*(c+d*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))^3),x,7,-(3*A*d*(2*c^2+2*c*d+d^2)-B*(2*c^3+4*c^2*d+7*c*d^2+2*d^3))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a*(c-d)*(c^2-d^2)^(5/2)*f)-1/2*d*(2*A*c-3*B*c+3*A*d-2*B*d)*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sin(e+f*x))^2)-(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^2)-1/2*d*(2*A*c^2-5*B*c^2+9*A*c*d-6*B*c*d+4*A*d^2-4*B*d^2)*cos(e+f*x)/(a*(c-d)^3*(c+d)^2*f*(c+d*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^2,x,3,1/2*d*(2*A*(3*c-2*d)*d+B*(6*c^2-12*c*d+7*d^2))*x/a^2+2/3*d*(A*(c^2+6*c*d-5*d^2)+B*(2*c^2-15*c*d+8*d^2))*cos(e+f*x)/(a^2*f)+1/6*d^2*(B*(4*c-21*d)+2*A*(c+6*d))*cos(e+f*x)*sin(e+f*x)/(a^2*f)-1/3*(2*B*(c-4*d)+A*(c+5*d))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a^2*f*(1+sin(e+f*x)))-1/3*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a+a*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^2,x,5,d*(2*B*(c-d)+A*d)*x/a^2+1/3*(A-4*B)*d^2*cos(e+f*x)/(a^2*f)-1/3*(c-d)*(2*B*(c-3*d)+A*(c+3*d))*cos(e+f*x)/(a^2*f*(1+sin(e+f*x)))-1/3*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^2,x,4,B*d*x/a^2-1/3*(A*c+2*B*c+2*A*d-5*B*d)*cos(e+f*x)/(a^2*f*(1+sin(e+f*x)))-1/3*(A-B)*(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^2)],
[(A+B*sin(e+f*x))/(a+a*sin(e+f*x))^2,x,2,-1/3*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^2)-1/3*(A+2*B)*cos(e+f*x)/(f*(a^2+a^2*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))),x,6,-1/3*(A*(c-4*d)+B*(2*c+d))*cos(e+f*x)/(a^2*(c-d)^2*f*(1+sin(e+f*x)))-1/3*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^2)-2*d*(B*c-A*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a^2*(c-d)^2*f*sqrt(c^2-d^2))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^2),x,7,-1/3*d*(A*(c^2-6*c*d-10*d^2)+B*(2*c^2+9*c*d+4*d^2))*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sin(e+f*x)))-1/3*(A*c+2*B*c-6*A*d+3*B*d)*cos(e+f*x)/(a^2*(c-d)^2*f*(1+sin(e+f*x))*(c+d*sin(e+f*x)))-1/3*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x)))+2*d*(A*d*(3*c+2*d)-B*(2*c^2+2*c*d+d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a^2*(c-d)^3*(c+d)*f*sqrt(c^2-d^2))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^3),x,8,-1/6*d*(A*(2*c^2-16*c*d-21*d^2)+B*(4*c^2+19*c*d+12*d^2))*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sin(e+f*x))^2)-1/3*(A*c+2*B*c-8*A*d+5*B*d)*cos(e+f*x)/(a^2*(c-d)^2*f*(1+sin(e+f*x))*(c+d*sin(e+f*x))^2)-1/3*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^2)-1/6*d*(A*(2*c^3-16*c^2*d-59*c*d^2-32*d^3)+B*(4*c^3+37*c^2*d+44*c*d^2+20*d^3))*cos(e+f*x)/(a^2*(c-d)^4*(c+d)^2*f*(c+d*sin(e+f*x)))+d*(A*d*(12*c^2+16*c*d+7*d^2)-B*(6*c^3+12*c^2*d+13*c*d^2+4*d^3))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a^2*(c-d)^4*(c+d)^2*f*sqrt(c^2-d^2))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^3,x,6,d^2*(3*B*(c-d)+A*d)*x/a^3+1/15*d^2*(3*B*(c-9*d)+A*(2*c+7*d))*cos(e+f*x)/(a^3*f)-1/15*(c-d)*(3*B*(c^2+6*c*d-15*d^2)+A*(2*c^2+7*c*d+15*d^2))*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))-1/15*(3*B*(c-3*d)+2*A*(c+2*d))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a*f*(a+a*sin(e+f*x))^2)-1/5*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a+a*sin(e+f*x))^3)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^3,x,5,B*d^2*x/a^3-1/15*(c-d)*(B*(3*c-7*d)+2*A*(c+d))*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^2)-1/15*(B*(3*c^2+14*c*d-29*d^2)+2*A*(c^2+3*c*d+2*d^2))*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))-1/5*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x))^3)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^3,x,4,-1/5*(A-B)*(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^3)-1/15*(2*A*c+3*B*c+3*A*d-8*B*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^2)-1/15*(2*A*c+3*B*c+3*A*d+7*B*d)*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))],
[(A+B*sin(e+f*x))/(a+a*sin(e+f*x))^3,x,3,-1/5*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^3)-1/15*(2*A+3*B)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^2)-1/15*(2*A+3*B)*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))),x,7,-1/5*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^3)-1/15*(2*A*c+3*B*c-7*A*d+2*B*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^2)-1/15*(B*(3*c^2-16*c*d-2*d^2)+A*(2*c^2-9*c*d+22*d^2))*cos(e+f*x)/((c-d)^3*f*(a^3+a^3*sin(e+f*x)))+2*d^2*(B*c-A*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a^3*(c-d)^3*f*sqrt(c^2-d^2))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^2),x,8,-1/15*d*(B*(3*c^3-23*c^2*d-63*c*d^2-22*d^3)+A*(2*c^3-12*c^2*d+43*c*d^2+72*d^3))*cos(e+f*x)/(a^3*(c-d)^4*(c+d)*f*(c+d*sin(e+f*x)))-1/5*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x)))-1/15*(2*A*c+3*B*c-9*A*d+4*B*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x)))-1/15*(B*(3*c^2-23*c*d-15*d^2)+A*(2*c^2-12*c*d+45*d^2))*cos(e+f*x)/((c-d)^3*f*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x)))-2*d^2*(A*d*(4*c+3*d)-B*(3*c^2+3*c*d+d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a^3*(c-d)^4*(c+d)*f*sqrt(c^2-d^2))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^3),x,9,-1/30*d*(3*B*(2*c^3-20*c^2*d-57*c*d^2-30*d^3)+A*(4*c^3-30*c^2*d+146*c*d^2+195*d^3))*cos(e+f*x)/(a^3*(c-d)^4*(c+d)*f*(c+d*sin(e+f*x))^2)-1/5*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^2)-1/15*(2*A*c+3*B*c-11*A*d+6*B*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^2)-1/15*(3*B*(c^2-10*c*d-12*d^2)+A*(2*c^2-15*c*d+76*d^2))*cos(e+f*x)/((c-d)^3*f*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x))^2)-1/30*d*(3*B*(2*c^4-20*c^3*d-119*c^2*d^2-130*c*d^3-48*d^4)+A*(4*c^4-30*c^3*d+142*c^2*d^2+525*c*d^3+304*d^4))*cos(e+f*x)/(a^3*(c-d)^5*(c+d)^2*f*(c+d*sin(e+f*x)))-d^2*(A*d*(20*c^2+30*c*d+13*d^2)-3*B*(4*c^3+8*c^2*d+7*c*d^2+2*d^3))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a^3*(c-d)^5*(c+d)^2*f*sqrt(c^2-d^2))],

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

# m>0
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3*sqrt(a+a*sin(e+f*x)),x,5,4/105*d*(c+d)*(B*c-9*A*d-8*B*d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(a*f)+4/315*a*(c+d)*(B*c-9*A*d-8*B*d)*(15*c^2+10*c*d+7*d^2)*cos(e+f*x)/(d*f*sqrt(a+a*sin(e+f*x)))+2/63*a*(B*c-9*A*d-8*B*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d*f*sqrt(a+a*sin(e+f*x)))-2/9*a*B*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d*f*sqrt(a+a*sin(e+f*x)))+8/315*(5*c-d)*(c+d)*(B*c-9*A*d-8*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2*sqrt(a+a*sin(e+f*x)),x,4,2/35*d*(B*c-7*A*d-6*B*d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(a*f)+2/105*a*(B*c-7*A*d-6*B*d)*(15*c^2+10*c*d+7*d^2)*cos(e+f*x)/(d*f*sqrt(a+a*sin(e+f*x)))-2/7*a*B*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d*f*sqrt(a+a*sin(e+f*x)))+4/105*(5*c-d)*(B*c-7*A*d-6*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)),x,4,-2/5*B*d*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(a*f)-2/15*a*(15*A*c+5*B*c+5*A*d+7*B*d)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/15*(5*B*c+5*A*d-2*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(A+B*sin(e+f*x))*sqrt(a+a*sin(e+f*x)),x,2,-2/3*a*(3*A+B)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/3*B*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(A+B*sin(e+f*x))*sqrt(a+a*sin(e+f*x))/(c+d*sin(e+f*x)),x,3,2*(B*c-A*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(a)/(d^(3/2)*f*sqrt(c+d))-2*a*B*cos(e+f*x)/(d*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*sqrt(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,3,-(A*d+B*(c+2*d))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(a)/(d^(3/2)*(c+d)^(3/2)*f)+a*(B*c-A*d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*sqrt(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,4,-1/4*(3*A*d+B*(c+4*d))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(a)/(d^(3/2)*(c+d)^(5/2)*f)+1/2*a*(B*c-A*d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^2*sqrt(a+a*sin(e+f*x)))-1/4*a*(3*A*d+B*(c+4*d))*cos(e+f*x)/(d*(c+d)^2*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3,x,6,4/1155*(c+d)*(11*A*(c-17*d)*d-3*B*(c^2-9*c*d+56*d^2))*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f+4/3465*a^2*(c+d)*(15*c^2+10*c*d+7*d^2)*(11*A*(c-17*d)*d-3*B*(c^2-9*c*d+56*d^2))*cos(e+f*x)/(d^2*f*sqrt(a+a*sin(e+f*x)))+2/693*a^2*(11*A*(c-17*d)*d-3*B*(c^2-9*c*d+56*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^2*f*sqrt(a+a*sin(e+f*x)))+2/99*a^2*(3*B*(c-4*d)-11*A*d)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d^2*f*sqrt(a+a*sin(e+f*x)))+8/3465*a*(5*c-d)*(c+d)*(11*A*(c-17*d)*d-3*B*(c^2-9*c*d+56*d^2))*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*f)-2/11*a*B*cos(e+f*x)*(c+d*sin(e+f*x))^4*sqrt(a+a*sin(e+f*x))/(d*f)],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2,x,5,2/105*(3*A*(c-13*d)*d-B*(c^2-7*c*d+34*d^2))*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f+2/315*a^2*(15*c^2+10*c*d+7*d^2)*(3*A*(c-13*d)*d-B*(c^2-7*c*d+34*d^2))*cos(e+f*x)/(d^2*f*sqrt(a+a*sin(e+f*x)))+2/63*a^2*(3*B*c-9*A*d-10*B*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^2*f*sqrt(a+a*sin(e+f*x)))+4/315*a*(5*c-d)*(3*A*(c-13*d)*d-B*(c^2-7*c*d+34*d^2))*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*f)-2/9*a*B*cos(e+f*x)*(c+d*sin(e+f*x))^3*sqrt(a+a*sin(e+f*x))/(d*f)],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))*(c+d*sin(e+f*x)),x,5,-2/35*(7*B*c+7*A*d-2*B*d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-2/7*B*d*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(a*f)-8/105*a^2*(35*A*c+21*B*c+21*A*d+19*B*d)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/105*a*(35*A*c+21*B*c+21*A*d+19*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x)),x,3,-2/5*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-8/15*a^2*(5*A+3*B)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/15*a*(5*A+3*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c+d*sin(e+f*x)),x,4,-2*a^(3/2)*(c-d)*(B*c-A*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(5/2)*f*sqrt(c+d))+2/3*a^2*(3*B*c-3*A*d-4*B*d)*cos(e+f*x)/(d^2*f*sqrt(a+a*sin(e+f*x)))-2/3*a*B*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*f)],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,4,-a^(3/2)*(A*d*(c+3*d)-B*(3*c^2+3*c*d-2*d^2))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(5/2)*(c+d)^(3/2)*f)-a^2*(3*B*c-A*d+2*B*d)*cos(e+f*x)/(d^2*(c+d)*f*sqrt(a+a*sin(e+f*x)))+a*(B*c-A*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,4,-1/4*a^(3/2)*(A*d*(c+7*d)+3*B*(c^2+3*c*d+4*d^2))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(5/2)*(c+d)^(5/2)*f)+1/4*a^2*(A*(c-5*d)*d+B*(3*c^2+5*c*d-4*d^2))*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))+1/2*a*(B*c-A*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^2)],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3,x,7,-4/15015*a*(c+d)*(13*A*d*(3*c^2-38*c*d+355*d^2)-B*(15*c^3-150*c^2*d+799*c*d^2-4184*d^3))*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(d*f)-2/13*a*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^4/(d*f)-4/45045*a^3*(c+d)*(15*c^2+10*c*d+7*d^2)*(13*A*d*(3*c^2-38*c*d+355*d^2)-B*(15*c^3-150*c^2*d+799*c*d^2-4184*d^3))*cos(e+f*x)/(d^3*f*sqrt(a+a*sin(e+f*x)))-2/9009*a^3*(13*A*d*(3*c^2-38*c*d+355*d^2)-B*(15*c^3-150*c^2*d+799*c*d^2-4184*d^3))*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^3*f*sqrt(a+a*sin(e+f*x)))-2/1287*a^3*(15*B*c^2-39*A*c*d-75*B*c*d+299*A*d^2+280*B*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d^3*f*sqrt(a+a*sin(e+f*x)))-8/45045*a^2*(5*c-d)*(c+d)*(13*A*d*(3*c^2-38*c*d+355*d^2)-B*(15*c^3-150*c^2*d+799*c*d^2-4184*d^3))*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d^2*f)+2/143*a^2*(5*B*c-13*A*d-16*B*d)*cos(e+f*x)*(c+d*sin(e+f*x))^4*sqrt(a+a*sin(e+f*x))/(d^2*f)],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2,x,6,-2/1155*a*(11*A*d*(c^2-10*c*d+73*d^2)-B*(5*c^3-40*c^2*d+169*c*d^2-710*d^3))*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(d*f)-2/11*a*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^3/(d*f)-2/3465*a^3*(15*c^2+10*c*d+7*d^2)*(11*A*d*(c^2-10*c*d+73*d^2)-B*(5*c^3-40*c^2*d+169*c*d^2-710*d^3))*cos(e+f*x)/(d^3*f*sqrt(a+a*sin(e+f*x)))+2/693*a^3*(11*A*(3*c-19*d)*d-B*(15*c^2-65*c*d+194*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^3*f*sqrt(a+a*sin(e+f*x)))-4/3465*a^2*(5*c-d)*(11*A*d*(c^2-10*c*d+73*d^2)-B*(5*c^3-40*c^2*d+169*c*d^2-710*d^3))*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d^2*f)+2/99*a^2*(5*B*c-11*A*d-14*B*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3*sqrt(a+a*sin(e+f*x))/(d^2*f)],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))*(c+d*sin(e+f*x)),x,6,-2/105*a*(21*A*c+15*B*c+15*A*d+13*B*d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-2/63*(9*B*c+9*A*d-2*B*d)*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/f-2/9*B*d*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*f)-64/315*a^3*(21*A*c+15*B*c+15*A*d+13*B*d)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-16/315*a^2*(21*A*c+15*B*c+15*A*d+13*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x)),x,4,-2/35*a*(7*A+5*B)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-2/7*B*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/f-64/105*a^3*(7*A+5*B)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-16/105*a^2*(7*A+5*B)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c+d*sin(e+f*x)),x,5,-2/5*a*B*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(d*f)+2*a^(5/2)*(c-d)^2*(B*c-A*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(7/2)*f*sqrt(c+d))+2/15*a^3*(5*A*(3*c-7*d)*d-B*(15*c^2-35*c*d+32*d^2))*cos(e+f*x)/(d^3*f*sqrt(a+a*sin(e+f*x)))+2/15*a^2*(5*B*c-5*A*d-8*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d^2*f)],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,5,a^(5/2)*(c-d)*(A*d*(3*c+5*d)-B*(5*c^2+5*c*d-2*d^2))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(7/2)*(c+d)^(3/2)*f)+a*(B*c-A*d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(d*(c+d)*f*(c+d*sin(e+f*x)))-1/3*a^3*(3*A*d*(3*c+d)-B*(15*c^2-5*c*d-14*d^2))*cos(e+f*x)/(d^3*(c+d)*f*sqrt(a+a*sin(e+f*x)))-1/3*a^2*(5*B*c-3*A*d+2*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d^2*(c+d)*f)],
[(a+a*sin(e+f*x))^(5/2)*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,5,-1/4*a^(5/2)*(A*d*(3*c^2+10*c*d+19*d^2)-B*(15*c^3+30*c^2*d+7*c*d^2-20*d^3))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(7/2)*(c+d)^(5/2)*f)+1/2*a*(B*c-A*d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(d*(c+d)*f*(c+d*sin(e+f*x))^2)+1/4*a^3*(3*A*d*(c+3*d)-B*(15*c^2+25*c*d+4*d^2))*cos(e+f*x)/(d^3*(c+d)^2*f*sqrt(a+a*sin(e+f*x)))-1/4*a^2*(A*d*(c+7*d)-B*(5*c^2+7*c*d-4*d^2))*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d^2*(c+d)^2*f*(c+d*sin(e+f*x)))],

# m<0
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3/sqrt(a+a*sin(e+f*x)),x,7,-(A-B)*(c-d)^3*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-4/105*(7*A*d*(21*c^2-12*c*d+7*d^2)+B*(36*c^3-63*c^2*d+144*c*d^2-37*d^3))*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/35*(6*B*c+7*A*d-B*d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*sqrt(a+a*sin(e+f*x)))-2/7*B*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*sqrt(a+a*sin(e+f*x)))-2/105*d*(7*A*(9*c-d)*d+B*(24*c^2-15*c*d+31*d^2))*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a*f)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2/sqrt(a+a*sin(e+f*x)),x,6,-(A-B)*(c-d)^2*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-4/15*(5*A*(3*c-d)*d+B*(6*c^2-7*c*d+7*d^2))*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/5*B*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*sqrt(a+a*sin(e+f*x)))-2/15*d*(4*B*c+5*A*d-B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a*f)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))/sqrt(a+a*sin(e+f*x)),x,5,-(A-B)*(c-d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-2/3*(3*B*c+3*A*d-2*B*d)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/3*B*d*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a*f)],
[(A+B*sin(e+f*x))/sqrt(a+a*sin(e+f*x)),x,3,-(A-B)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-2*B*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x))),x,5,-(A-B)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/((c-d)*f*sqrt(a))-2*(B*c-A*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/((c-d)*f*sqrt(a)*sqrt(d)*sqrt(c+d))],
[(A+B*sin(e+f*x))/((c+d*sin(e+f*x))^2*sqrt(a+a*sin(e+f*x))),x,6,-(A-B)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/((c-d)^2*f*sqrt(a))+(A*d*(3*c+d)-B*(c^2+c*d+2*d^2))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/((c-d)^2*(c+d)^(3/2)*f*sqrt(a)*sqrt(d))-(B*c-A*d)*cos(e+f*x)/((c^2-d^2)*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((c+d*sin(e+f*x))^3*sqrt(a+a*sin(e+f*x))),x,7,-(A-B)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/((c-d)^3*f*sqrt(a))+1/4*(A*d*(15*c^2+10*c*d+7*d^2)-B*(3*c^3+6*c^2*d+19*c*d^2+4*d^3))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/((c-d)^3*(c+d)^(5/2)*f*sqrt(a)*sqrt(d))-1/2*(B*c-A*d)*cos(e+f*x)/((c^2-d^2)*f*(c+d*sin(e+f*x))^2*sqrt(a+a*sin(e+f*x)))+1/4*(A*d*(7*c+d)-B*(3*c^2+c*d+4*d^2))*cos(e+f*x)/((c^2-d^2)^2*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^(3/2),x,7,-1/2*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(c-d)^2*(3*B*(c-5*d)+A*(c+11*d))*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*f*sqrt(2))+1/15*d*(15*A*c^2-99*B*c^2-120*A*c*d+168*B*c*d+65*A*d^2-93*B*d^2)*cos(e+f*x)/(a*f*sqrt(a+a*sin(e+f*x)))+1/10*(5*A-9*B)*d*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a*f*sqrt(a+a*sin(e+f*x)))+1/30*d^2*(15*A*c-51*B*c-35*A*d+39*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a^2*f)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^(3/2),x,6,-1/2*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(c-d)*(A*c+3*B*c+7*A*d-11*B*d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*f*sqrt(2))+1/3*d*(3*A*c-15*B*c-9*A*d+13*B*d)*cos(e+f*x)/(a*f*sqrt(a+a*sin(e+f*x)))+1/6*(3*A-7*B)*d^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a^2*f)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^(3/2),x,5,-1/2*(A-B)*(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(A*c+3*B*c+3*A*d-7*B*d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*f*sqrt(2))-2*B*d*cos(e+f*x)/(a*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))/(a+a*sin(e+f*x))^(3/2),x,3,-1/2*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(A+3*B)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*f*sqrt(2))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))),x,6,-1/2*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(3/2))-1/2*(A*(c-5*d)+B*(3*c+d))*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*(c-d)^2*f*sqrt(2))+2*(B*c-A*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(d)/(a^(3/2)*(c-d)^2*f*sqrt(c+d))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^2),x,7,-1/2*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x)))-1/2*(A*c+3*B*c-9*A*d+5*B*d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*(c-d)^3*f*sqrt(2))-(A*d*(5*c+3*d)-B*(3*c^2+3*c*d+2*d^2))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(d)/(a^(3/2)*(c-d)^3*(c+d)^(3/2)*f)+1/2*d*(B*(3*c+d)-A*(c+3*d))*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^3),x,8,-1/2*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^2)-1/2*(A*(c-13*d)+3*B*(c+3*d))*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*(c-d)^4*f*sqrt(2))-1/4*(A*d*(35*c^2+42*c*d+19*d^2)-3*B*(5*c^3+10*c^2*d+13*c*d^2+4*d^3))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(d)/(a^(3/2)*(c-d)^4*(c+d)^(5/2)*f)+1/2*d*(B*(2*c+d)-A*(c+2*d))*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sin(e+f*x))^2*sqrt(a+a*sin(e+f*x)))+1/4*d*(3*B*(3*c^2+3*c*d+2*d^2)-A*(2*c^2+15*c*d+7*d^2))*cos(e+f*x)/(a*(c-d)^3*(c+d)^2*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^(5/2),x,7,-1/16*(3*A*c+5*B*c+9*A*d-17*B*d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a*f*(a+a*sin(e+f*x))^(3/2))-1/4*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(c-d)*(B*(5*c^2+62*c*d-163*d^2)+3*A*(c^2+6*c*d+25*d^2))*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*f*sqrt(2))+1/24*d*(A*(9*c^2+36*c*d-93*d^2)+B*(15*c^2-228*c*d+197*d^2))*cos(e+f*x)/(a^2*f*sqrt(a+a*sin(e+f*x)))+1/48*d^2*(9*A*c+15*B*c+39*A*d-95*B*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a^3*f)],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^(5/2),x,6,-1/16*(c-d)*(3*A*c+5*B*c+5*A*d-13*B*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/4*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(B*(5*c^2+38*c*d-75*d^2)+A*(3*c^2+10*c*d+19*d^2))*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*f*sqrt(2))+1/4*(A-9*B)*d^2*cos(e+f*x)/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^(5/2),x,5,-1/4*(A-B)*(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(3*A*c+5*B*c+5*A*d-13*B*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/16*(3*A*c+5*B*c+5*A*d+19*B*d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*f*sqrt(2))],
[(A+B*sin(e+f*x))/(a+a*sin(e+f*x))^(5/2),x,4,-1/4*(A-B)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(3*A+5*B)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/16*(3*A+5*B)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*f*sqrt(2))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))),x,7,-1/4*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(5/2))-1/16*(3*A*c+5*B*c-11*A*d+3*B*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^(3/2))-1/16*(B*(5*c^2-34*c*d-3*d^2)+A*(3*c^2-14*c*d+43*d^2))*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*(c-d)^3*f*sqrt(2))-2*d^(3/2)*(B*c-A*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*(c-d)^3*f*sqrt(c+d))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^2),x,8,d^(3/2)*(A*d*(7*c+5*d)-B*(5*c^2+5*c*d+2*d^2))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*(c-d)^4*(c+d)^(3/2)*f)-1/4*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x)))-1/16*(3*A*c+5*B*c-15*A*d+7*B*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x)))-1/16*(B*(5*c^2-58*c*d-43*d^2)+A*(3*c^2-22*c*d+115*d^2))*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*(c-d)^4*f*sqrt(2))-1/16*d*(A*(3*c^2-16*c*d-35*d^2)+B*(5*c^2+32*c*d+11*d^2))*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^3),x,9,1/4*d^(3/2)*(3*A*d*(21*c^2+30*c*d+13*d^2)-B*(35*c^3+70*c^2*d+67*c*d^2+20*d^3))*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*(c-d)^5*(c+d)^(5/2)*f)-1/4*(A-B)*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^2)-1/16*(3*A*c+5*B*c-19*A*d+11*B*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^2)-1/16*(B*(5*c^2-82*c*d-115*d^2)+3*A*(c^2-10*c*d+73*d^2))*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*(c-d)^5*f*sqrt(2))-1/16*d*(A*(3*c^2-20*c*d-31*d^2)+B*(5*c^2+28*c*d+15*d^2))*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sin(e+f*x))^2*sqrt(a+a*sin(e+f*x)))-1/16*d*(3*A*(c^3-7*c^2*d-37*c*d^2-21*d^3)+B*(5*c^3+73*c^2*d+79*c*d^2+35*d^3))*cos(e+f*x)/(a^2*(c-d)^4*(c+d)^2*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],

# Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]) with m and/or n symbolic
[(a+a*sin(e+f*x))^2*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,7,-8*a^2*B*AppellF1(1/2,-5/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))-4*a^2*(A-B)*AppellF1(1/2,-3/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[(a+a*sin(e+f*x))*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,8,-4*a*B*AppellF1(1/2,-3/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))-2*a*(A-B)*AppellF1(1/2,-1/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x)),x,7,-(A-B)*AppellF1(1/2,3/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n/(a*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(2)*sqrt(1+sin(e+f*x)))-B*AppellF1(1/2,1/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(a*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^2,x,7,-B*AppellF1(1/2,3/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n/(a^2*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(2)*sqrt(1+sin(e+f*x)))-1/2*(A-B)*AppellF1(1/2,5/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n/(a^2*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(2)*sqrt(1+sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,11,-2*a^2*(A-B)*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)/(d*f*(3+2*n)*sqrt(a+a*sin(e+f*x)))+2*a^2*B*(3*c-d*(11+4*n))*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)/(d^2*f*(15+16*n+4*n^2)*sqrt(a+a*sin(e+f*x)))+2*a^2*(A-B)*(c-d*(5+4*n))*cos(e+f*x)*hypergeom([1/2,-n],[3/2],d*(1-sin(e+f*x))/(c+d))*(c+d*sin(e+f*x))^n/(d*f*(3+2*n)*((c+d*sin(e+f*x))/(c+d))^n*sqrt(a+a*sin(e+f*x)))-2*a^2*B*(3*c^2-2*c*d*(7+4*n)+d^2*(43+56*n+16*n^2))*cos(e+f*x)*hypergeom([1/2,-n],[3/2],d*(1-sin(e+f*x))/(c+d))*(c+d*sin(e+f*x))^n/(d^2*f*(15+16*n+4*n^2)*((c+d*sin(e+f*x))/(c+d))^n*sqrt(a+a*sin(e+f*x)))-2*a*B*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)*sqrt(a+a*sin(e+f*x))/(d*f*(5+2*n))],
[(a+a*sin(e+f*x))^(1/2)*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,4,-2*a*B*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)/(d*f*(3+2*n)*sqrt(a+a*sin(e+f*x)))-2*a*(A*d*(3+2*n)-B*(c-2*d*(1+n)))*cos(e+f*x)*hypergeom([1/2,-n],[3/2],d*(1-sin(e+f*x))/(c+d))*(c+d*sin(e+f*x))^n/(d*f*(3+2*n)*((c+d*sin(e+f*x))/(c+d))^n*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^(1/2),x,7,-2*B*cos(e+f*x)*hypergeom([1/2,-n],[3/2],d*(1-sin(e+f*x))/(c+d))*(c+d*sin(e+f*x))^n/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(a+a*sin(e+f*x)))-(A-B)*AppellF1(1+n,1/2,1,2+n,(c+d*sin(e+f*x))/(c+d),(c+d*sin(e+f*x))/(c-d))*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)*sqrt(d*(1-sin(e+f*x))/(c+d))/((c-d)*f*(1+n)*(1-sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^(3/2),x,7,-B*AppellF1(1+n,1/2,1,2+n,(c+d*sin(e+f*x))/(c+d),(c+d*sin(e+f*x))/(c-d))*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)*sqrt(d*(1-sin(e+f*x))/(c+d))/(a*(c-d)*f*(1+n)*(1-sin(e+f*x))*sqrt(a+a*sin(e+f*x)))+(A-B)*d*AppellF1(1+n,1/2,2,2+n,(c+d*sin(e+f*x))/(c+d),(c+d*sin(e+f*x))/(c-d))*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)*sqrt(d*(1-sin(e+f*x))/(c+d))/((c-d)^2*f*(1+n)*(a-a*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^2,x,6,(d*(A*d*(3+m)+B*(2*c+d*m))-2*(2+m)*(A*c*d*(3+m)+B*(c^2+d^2+c*d*m)))*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+m)*(2+m)*(3+m))-2^(1/2+m)*(A*(3+m)*(2*c*d*m*(2+m)+d^2*(1+m+m^2)+c^2*(2+3*m+m^2))+B*(d^2*m*(5+3*m+m^2)+c^2*m*(6+5*m+m^2)+2*c*d*(3+4*m+4*m^2+m^3)))*cos(e+f*x)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(f*(3+m)*(2+3*m+m^2))-d*(A*d*(3+m)+B*(2*c+d*m))*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(2+m)*(3+m))-B*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^2/(f*(3+m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c+d*sin(e+f*x)),x,5,(B*d-(B*c+A*d)*(2+m))*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+m)*(2+m))-2^(1/2+m)*(A*(2+m)*(c+c*m+d*m)+B*(c*m*(2+m)+d*(1+m+m^2)))*cos(e+f*x)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(f*(2+3*m+m^2))-B*d*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(2+m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,-B*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+m))-2^(1/2+m)*(A+A*m+B*m)*cos(e+f*x)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(f*(1+m))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c+d*sin(e+f*x)),x,6,-2^(1/2+m)*B*cos(e+f*x)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(d*f)-(B*c-A*d)*AppellF1(1/2+m,1/2,1,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(2)/((c-d)*d*f*(1+2*m)*sqrt(1-sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,7,2^(1/2+m)*(B*c-A*d)*m*cos(e+f*x)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(d*(c^2-d^2)*f)-(B*c-A*d)*cos(e+f*x)*(a+a*sin(e+f*x))^m/((c^2-d^2)*f*(c+d*sin(e+f*x)))+(A*d*(c*(1-m)-d*m)-B*(d^2-c^2*m-c*d*m))*AppellF1(1/2+m,1/2,1,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(2)/((c-d)^2*d*(c+d)*f*(1+2*m)*sqrt(1-sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,8,-2^(-1/2+m)*m*(A*d*(c*(3-m)-d*m)-B*(2*d^2+c^2*(1-m)-c*d*m))*cos(e+f*x)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(d*(c^2-d^2)^2*f)-1/2*(B*c-A*d)*cos(e+f*x)*(a+a*sin(e+f*x))^m/((c^2-d^2)*f*(c+d*sin(e+f*x))^2)+1/2*(A*d*(c*(3-m)-d*m)-B*(2*d^2+c^2*(1-m)-c*d*m))*cos(e+f*x)*(a+a*sin(e+f*x))^m/((c^2-d^2)^2*f*(c+d*sin(e+f*x)))+(B*(2*d^3*m+c^3*(1-m)*m+2*c^2*d*(1-m)*m-c*d^2*(3-3*m+m^2))-A*d*(2*c*d*(2-m)*m-c^2*(2-3*m+m^2)-d^2*(1-m+m^2)))*AppellF1(1/2+m,1/2,1,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^m/((c-d)^3*d*(c+d)^2*f*(1+2*m)*sqrt(2)*sqrt(1-sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2),x,9,(A-B)*(c-d)*AppellF1(1/2+m,1/2,-3/2,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(2)*sqrt(c+d*sin(e+f*x))/(f*(1+2*m)*sqrt(1-sin(e+f*x))*sqrt((c+d*sin(e+f*x))/(c-d)))+B*(c-d)*AppellF1(3/2+m,1/2,-3/2,5/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*sqrt(2)*sqrt(c+d*sin(e+f*x))/(a*f*(3+2*m)*sqrt(1-sin(e+f*x))*sqrt((c+d*sin(e+f*x))/(c-d)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^(1/2),x,9,(A-B)*AppellF1(1/2+m,1/2,-1/2,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(2)*sqrt(c+d*sin(e+f*x))/(f*(1+2*m)*sqrt(1-sin(e+f*x))*sqrt((c+d*sin(e+f*x))/(c-d)))+B*AppellF1(3/2+m,1/2,-1/2,5/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*sqrt(2)*sqrt(c+d*sin(e+f*x))/(a*f*(3+2*m)*sqrt(1-sin(e+f*x))*sqrt((c+d*sin(e+f*x))/(c-d)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^(1/2),x,9,(A-B)*AppellF1(1/2+m,1/2,1/2,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(2)*sqrt((c+d*sin(e+f*x))/(c-d))/(f*(1+2*m)*sqrt(1-sin(e+f*x))*sqrt(c+d*sin(e+f*x)))+B*AppellF1(3/2+m,1/2,1/2,5/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*sqrt(2)*sqrt((c+d*sin(e+f*x))/(c-d))/(a*f*(3+2*m)*sqrt(1-sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^(3/2),x,9,(A-B)*AppellF1(1/2+m,1/2,3/2,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(2)*sqrt((c+d*sin(e+f*x))/(c-d))/((c-d)*f*(1+2*m)*sqrt(1-sin(e+f*x))*sqrt(c+d*sin(e+f*x)))+B*AppellF1(3/2+m,1/2,3/2,5/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*sqrt(2)*sqrt((c+d*sin(e+f*x))/(c-d))/(a*(c-d)*f*(3+2*m)*sqrt(1-sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,9,(A-B)*AppellF1(1/2+m,1/2,-n,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n*sqrt(2)/(f*(1+2*m)*((c+d*sin(e+f*x))/(c-d))^n*sqrt(1-sin(e+f*x)))+B*AppellF1(3/2+m,1/2,-n,5/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c+d*sin(e+f*x))^n*sqrt(2)/(a*f*(3+2*m)*((c+d*sin(e+f*x))/(c-d))^n*sqrt(1-sin(e+f*x)))],
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^(-1-m),x,7,-2^(1/2+m)*a*(A-B)*cos(e+f*x)*hypergeom([1/2,1/2-m],[3/2],1/2*(c-d)*(1-sin(e+f*x))/(c+d*sin(e+f*x)))*(a+a*sin(e+f*x))^(-1+m)*((c+d)*(1+sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2-m)/((c+d)*f*(c+d*sin(e+f*x))^m)+B*AppellF1(3/2+m,1/2,1+m,5/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*((c+d*sin(e+f*x))/(c-d))^m*sqrt(2)/(a*(c-d)*f*(3+2*m)*(c+d*sin(e+f*x))^m*sqrt(1-sin(e+f*x)))],
[(a-a*sin(e+f*x))*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x,4,2*AppellF1(1/2+m,-1/2,-n,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*sec(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c+d*sin(e+f*x))^n*sqrt(2)*sqrt(1-sin(e+f*x))/(f*(1+2*m)*((c+d*sin(e+f*x))/(c-d))^n)],
[(a-a*sin(e+f*x))*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^(-1-m),x,4,2*AppellF1(1/2+m,-1/2,1+m,3/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*sec(e+f*x)*(a+a*sin(e+f*x))^(1+m)*((c+d*sin(e+f*x))/(c-d))^m*sqrt(2)*sqrt(1-sin(e+f*x))/((c-d)*f*(1+2*m)*(c+d*sin(e+f*x))^m)],
[(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^(-2-m)*(d-(c-d)*m+(c+(c-d)*m)*sin(e+f*x)),x,1,-cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^(-1-m)/f],
[(a-a*sin(e+f*x))^m*(c+d*sin(e+f*x))^(-2-m)*(d+(c+d)*m+(c+(c+d)*m)*sin(e+f*x)),x,1,-cos(e+f*x)*(a-a*sin(e+f*x))^m*(c+d*sin(e+f*x))^(-1-m)/f],

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

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

# m>0
[(a+b*sin(e+f*x))^2*(A+B*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,6,-b*(2*b*B*c-A*b*d-2*a*B*d)*x/d^3-2*(b*c-a*d)*(a*d^2*(A*c-B*d)-b*(2*B*c^3-A*c^2*d-3*B*c*d^2+2*A*d^3))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^3*(c^2-d^2)^(3/2)*f)-b^2*B*cos(e+f*x)/(d^2*f)-(b*c-a*d)^2*(B*c-A*d)*cos(e+f*x)/(d^2*(c^2-d^2)*f*(c+d*sin(e+f*x)))],

# m<0

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

# Integrands of the form (a+b Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2)  (A+B Sin[e+f x])
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2)/(a+b*sin(e+f*x))^(3/2),x,7,(c-d)*(2*A*b^2*c-2*a*b*B*c-2*a*A*b*d+3*a^2*B*d-b^2*B*d)*EllipticE(sqrt(a+b)*sqrt(c+d*sin(e+f*x))/(sqrt(c+d)*sqrt(a+b*sin(e+f*x))),sqrt((a-b)*(c+d)/((a+b)*(c-d))))*sec(e+f*x)*(a+b*sin(e+f*x))*sqrt(c+d)*sqrt(-(b*c-a*d)*(1-sin(e+f*x))/((c+d)*(a+b*sin(e+f*x))))*sqrt((b*c-a*d)*(1+sin(e+f*x))/((c-d)*(a+b*sin(e+f*x))))/((a-b)*b^2*(b*c-a*d)*f*sqrt(a+b))+(3*b*B*c+2*A*b*d-3*a*B*d)*EllipticPi(sqrt(a+b)*sqrt(c+d*sin(e+f*x))/(sqrt(c+d)*sqrt(a+b*sin(e+f*x))),b*(c+d)/((a+b)*d),sqrt((a-b)*(c+d)/((a+b)*(c-d))))*sec(e+f*x)*(a+b*sin(e+f*x))*sqrt(c+d)*sqrt(-(b*c-a*d)*(1-sin(e+f*x))/((c+d)*(a+b*sin(e+f*x))))*sqrt((b*c-a*d)*(1+sin(e+f*x))/((c-d)*(a+b*sin(e+f*x))))/(b^3*f*sqrt(a+b))+(2*A*b*(b*(c-2*d)+a*d)-B*(3*a^2*d-6*a*b*d+b^2*(2*c+d)))*EllipticF(sqrt(c+d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(c+d*sin(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*sec(e+f*x)*(c+d*sin(e+f*x))*sqrt(a+b)*sqrt((b*c-a*d)*(1-sin(e+f*x))/((a+b)*(c+d*sin(e+f*x))))*sqrt(-(b*c-a*d)*(1+sin(e+f*x))/((a-b)*(c+d*sin(e+f*x))))/((a-b)*b^3*f*sqrt(c+d))+2*(A*b-a*B)*(b*c-a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*sqrt(a+b*sin(e+f*x)))-(2*A*b*(b*c-a*d)-B*(2*a*b*c-3*a^2*d+b^2*d))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*sqrt(a+b*sin(e+f*x)))],
[(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x))^(3/2),x,5,2*(A*b-a*B)*(c-d)*EllipticE(sqrt(a+b)*sqrt(c+d*sin(e+f*x))/(sqrt(c+d)*sqrt(a+b*sin(e+f*x))),sqrt((a-b)*(c+d)/((a+b)*(c-d))))*sec(e+f*x)*(a+b*sin(e+f*x))*sqrt(c+d)*sqrt(-(b*c-a*d)*(1-sin(e+f*x))/((c+d)*(a+b*sin(e+f*x))))*sqrt((b*c-a*d)*(1+sin(e+f*x))/((c-d)*(a+b*sin(e+f*x))))/((a-b)*b*(b*c-a*d)*f*sqrt(a+b))+2*(A*b-a*B)*(c-d)*EllipticF(sqrt(c+d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(c+d*sin(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*sec(e+f*x)*(c+d*sin(e+f*x))*sqrt(a+b)*sqrt((b*c-a*d)*(1-sin(e+f*x))/((a+b)*(c+d*sin(e+f*x))))*sqrt(-(b*c-a*d)*(1+sin(e+f*x))/((a-b)*(c+d*sin(e+f*x))))/((a-b)*b*(b*c-a*d)*f*sqrt(c+d))+2*B*EllipticPi(sqrt(c+d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(c+d*sin(e+f*x))),(a+b)*d/(b*(c+d)),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*sec(e+f*x)*(c+d*sin(e+f*x))*sqrt(a+b)*sqrt((b*c-a*d)*(1-sin(e+f*x))/((a+b)*(c+d*sin(e+f*x))))*sqrt(-(b*c-a*d)*(1+sin(e+f*x))/((a-b)*(c+d*sin(e+f*x))))/(b^2*f*sqrt(c+d))],
[(A+B*sin(e+f*x))/((a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(1/2)),x,3,2*(A*b-a*B)*(c-d)*EllipticE(sqrt(a+b)*sqrt(c+d*sin(e+f*x))/(sqrt(c+d)*sqrt(a+b*sin(e+f*x))),sqrt((a-b)*(c+d)/((a+b)*(c-d))))*sec(e+f*x)*(a+b*sin(e+f*x))*sqrt(c+d)*sqrt(-(b*c-a*d)*(1-sin(e+f*x))/((c+d)*(a+b*sin(e+f*x))))*sqrt((b*c-a*d)*(1+sin(e+f*x))/((c-d)*(a+b*sin(e+f*x))))/((a-b)*(b*c-a*d)^2*f*sqrt(a+b))+2*(A-B)*EllipticF(sqrt(c+d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(c+d*sin(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*sec(e+f*x)*(c+d*sin(e+f*x))*sqrt(a+b)*sqrt((b*c-a*d)*(1-sin(e+f*x))/((a+b)*(c+d*sin(e+f*x))))*sqrt(-(b*c-a*d)*(1+sin(e+f*x))/((a-b)*(c+d*sin(e+f*x))))/((a-b)*(b*c-a*d)*f*sqrt(c+d))],
[(A+B*sin(e+f*x))/((a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(3/2)),x,4,-2*(A*(a^2*d^2+b^2*(c^2-2*d^2))-B*(a^2*c*d-b^2*c*d+a*b*(c^2-d^2)))*EllipticE(sqrt(c+d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(c+d*sin(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*sec(e+f*x)*(c+d*sin(e+f*x))*sqrt((b*c-a*d)*(1-sin(e+f*x))/((a+b)*(c+d*sin(e+f*x))))*sqrt(-(b*c-a*d)*(1+sin(e+f*x))/((a-b)*(c+d*sin(e+f*x))))/((c-d)*(b*c-a*d)^3*f*sqrt(a+b)*sqrt(c+d))+2*(A*b*c+b*B*c-a*A*d-2*A*b*d+a*B*d)*EllipticF(sqrt(c+d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(c+d*sin(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*sec(e+f*x)*(c+d*sin(e+f*x))*sqrt((b*c-a*d)*(1-sin(e+f*x))/((a+b)*(c+d*sin(e+f*x))))*sqrt(-(b*c-a*d)*(1+sin(e+f*x))/((a-b)*(c+d*sin(e+f*x))))/((c-d)*(b*c-a*d)^2*f*sqrt(a+b)*sqrt(c+d))+2*b*(A*b-a*B)*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(A+B*sin(e+f*x))/((a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(5/2)),x,5,2*b*(A*b-a*B)*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+b*sin(e+f*x)))+2/3*d*(A*(a^2*d^2+b^2*(3*c^2-4*d^2))-B*(a^2*c*d-b^2*c*d+3*a*b*(c^2-d^2)))*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/((a^2-b^2)*(b*c-a*d)^2*(c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))+2/3*(B*(2*a^2*b*c*d*(3*c^2-d^2)-2*b^3*c*d*(3*c^2-d^2)-a^3*d^2*(c^2+3*d^2)+a*b^2*(3*c^4-5*c^2*d^2+6*d^4))+A*(4*a^3*c*d^3-4*a*b^2*c*d^3-a^2*b*d^2*(9*c^2-5*d^2)-b^3*(3*c^4-15*c^2*d^2+8*d^4)))*EllipticE(sqrt(c+d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(c+d*sin(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*sec(e+f*x)*(c+d*sin(e+f*x))*sqrt((b*c-a*d)*(1-sin(e+f*x))/((a+b)*(c+d*sin(e+f*x))))*sqrt(-(b*c-a*d)*(1+sin(e+f*x))/((a-b)*(c+d*sin(e+f*x))))/((c-d)^2*(c+d)^(3/2)*(b*c-a*d)^4*f*sqrt(a+b))-2/3*(B*(a^2*d^2*(c+3*d)-b^2*c*(3*c^2+3*c*d-2*d^2)-6*a*b*d*(c^2-d^2))-A*(a^2*d^2*(3*c+d)-6*a*b*d*(c^2-d^2)+b^2*(3*c^3-9*c^2*d-6*c*d^2+8*d^3)))*EllipticF(sqrt(c+d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(c+d*sin(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*sec(e+f*x)*(c+d*sin(e+f*x))*sqrt((b*c-a*d)*(1-sin(e+f*x))/((a+b)*(c+d*sin(e+f*x))))*sqrt(-(b*c-a*d)*(1+sin(e+f*x))/((a-b)*(c+d*sin(e+f*x))))/((c-d)^2*(c+d)^(3/2)*(b*c-a*d)^3*f*sqrt(a+b))],

# Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]) with m and/or n symbolic
[(a+b*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,0,Unintegrable((a+b*sin(e+f*x))^m*(A+B*sin(e+f*x))*(c+d*sin(e+f*x))^n,x)]]:
