# Maple integration test file: "4 Trig functions\4.1 Sine\4.1.2.1 (a+b sin)^m (c+d sin)^n.txt"

lst:=[

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

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

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

# n>0
[sin(e+f*x)^3*(a+a*sin(e+f*x))^2,x,9,3/4*a^2*x-2*a^2*cos(e+f*x)/f+a^2*cos(e+f*x)^3/f-1/5*a^2*cos(e+f*x)^5/f-3/4*a^2*cos(e+f*x)*sin(e+f*x)/f-1/2*a^2*cos(e+f*x)*sin(e+f*x)^3/f],
[sin(e+f*x)^3*(a+a*sin(e+f*x))^3,x,13,23/16*a^3*x-4*a^3*cos(e+f*x)/f+7/3*a^3*cos(e+f*x)^3/f-3/5*a^3*cos(e+f*x)^5/f-23/16*a^3*cos(e+f*x)*sin(e+f*x)/f-23/24*a^3*cos(e+f*x)*sin(e+f*x)^3/f-1/6*a^3*cos(e+f*x)*sin(e+f*x)^5/f],

# n<0
[sin(x)^4/(a+a*sin(x)),x,6,-3/2*x/a-4*cos(x)/a+4/3*cos(x)^3/a+3/2*cos(x)*sin(x)/a+cos(x)*sin(x)^3/(a+a*sin(x))],
[sin(x)^3/(a+a*sin(x)),x,2,3/2*x/a+2*cos(x)/a-3/2*cos(x)*sin(x)/a+cos(x)*sin(x)^2/(a+a*sin(x))],
[sin(x)^2/(a+a*sin(x)),x,4,-x/a-cos(x)/a-cos(x)/(a*(1+sin(x)))],
[sin(x)/(a+a*sin(x)),x,2,x/a+cos(x)/(a+a*sin(x))],
[1/(a+a*sin(x)),x,1,-cos(x)/(a+a*sin(x))],
[csc(x)/(a+a*sin(x)),x,3,-arctanh(cos(x))/a+cos(x)/(a+a*sin(x))],
[csc(x)^2/(a+a*sin(x)),x,5,arctanh(cos(x))/a-2*cot(x)/a+cot(x)/(a+a*sin(x))],
[csc(x)^3/(a+a*sin(x)),x,6,-3/2*arctanh(cos(x))/a+2*cot(x)/a-3/2*cot(x)*csc(x)/a+cot(x)*csc(x)/(a+a*sin(x))],
[csc(x)^4/(a+a*sin(x)),x,6,3/2*arctanh(cos(x))/a-4*cot(x)/a-4/3*cot(x)^3/a+3/2*cot(x)*csc(x)/a+cot(x)*csc(x)^2/(a+a*sin(x))],
[sin(x)^4/(a+a*sin(x))^2,x,3,7/2*x/a^2+16/3*cos(x)/a^2-7/2*cos(x)*sin(x)/a^2+8/3*cos(x)*sin(x)^2/(a^2*(1+sin(x)))+1/3*cos(x)*sin(x)^3/(a+a*sin(x))^2],
[sin(x)^3/(a+a*sin(x))^2,x,6,-2*x/a^2-4/3*cos(x)/a^2-2*cos(x)/(a^2*(1+sin(x)))+1/3*cos(x)*sin(x)^2/(a+a*sin(x))^2],
[sin(x)^2/(a+a*sin(x))^2,x,3,x/a^2+5/3*cos(x)/(a^2*(1+sin(x)))-1/3*cos(x)/(a+a*sin(x))^2],
[sin(x)/(a+a*sin(x))^2,x,2,1/3*cos(x)/(a+a*sin(x))^2-2/3*cos(x)/(a^2+a^2*sin(x))],
[1/(a+a*sin(x))^2,x,2,-1/3*cos(x)/(a+a*sin(x))^2-1/3*cos(x)/(a^2+a^2*sin(x))],
[csc(x)/(a+a*sin(x))^2,x,4,-arctanh(cos(x))/a^2+4/3*cos(x)/(a^2*(1+sin(x)))+1/3*cos(x)/(a+a*sin(x))^2],
[csc(x)^2/(a+a*sin(x))^2,x,6,2*arctanh(cos(x))/a^2-10/3*cot(x)/a^2+2*cot(x)/(a^2*(1+sin(x)))+1/3*cot(x)/(a+a*sin(x))^2],
[csc(x)^3/(a+a*sin(x))^2,x,7,-7/2*arctanh(cos(x))/a^2+16/3*cot(x)/a^2-7/2*cot(x)*csc(x)/a^2+8/3*cot(x)*csc(x)/(a^2*(1+sin(x)))+1/3*cot(x)*csc(x)/(a+a*sin(x))^2],
[csc(x)^4/(a+a*sin(x))^2,x,7,5*arctanh(cos(x))/a^2-4*cot(x)/a^2-1/3*cot(x)^3/a^2+cot(x)*csc(x)/a^2-1/3*cos(x)/(a^2*(1+sin(x))^2)-13/3*cos(x)/(a^2*(1+sin(x))),5*arctanh(cos(x))/a^2-12*cot(x)/a^2-4*cot(x)^3/a^2+5*cot(x)*csc(x)/a^2+10/3*cot(x)*csc(x)^2/(a^2*(1+sin(x)))+1/3*cot(x)*csc(x)^2/(a+a*sin(x))^2],
[sin(x)^6/(a+a*sin(x))^3,x,8,-23/2*x/a^3-136/5*cos(x)/a^3+136/15*cos(x)^3/a^3+23/2*cos(x)*sin(x)/a^3+1/5*cos(x)*sin(x)^5/(a+a*sin(x))^3+13/15*cos(x)*sin(x)^4/(a*(a+a*sin(x))^2)+23/3*cos(x)*sin(x)^3/(a^3+a^3*sin(x))],
[sin(x)^5/(a+a*sin(x))^3,x,4,13/2*x/a^3+152/15*cos(x)/a^3-13/2*cos(x)*sin(x)/a^3+1/5*cos(x)*sin(x)^4/(a+a*sin(x))^3+11/15*cos(x)*sin(x)^3/(a*(a+a*sin(x))^2)+76/15*cos(x)*sin(x)^2/(a^3+a^3*sin(x))],
[sin(x)^4/(a+a*sin(x))^3,x,7,-3*x/a^3-9/5*cos(x)/a^3+1/5*cos(x)*sin(x)^3/(a+a*sin(x))^3+3/5*cos(x)*sin(x)^2/(a*(a+a*sin(x))^2)-3*cos(x)/(a^3+a^3*sin(x))],
[sin(x)^3/(a+a*sin(x))^3,x,5,x/a^3+1/5*cos(x)*sin(x)^2/(a+a*sin(x))^3-7/15*cos(x)/(a*(a+a*sin(x))^2)+29/15*cos(x)/(a^3+a^3*sin(x))],
[sin(x)^2/(a+a*sin(x))^3,x,3,-1/5*cos(x)/(a+a*sin(x))^3+8/15*cos(x)/(a*(a+a*sin(x))^2)-7/15*cos(x)/(a^3+a^3*sin(x))],
[sin(x)/(a+a*sin(x))^3,x,3,1/5*cos(x)/(a+a*sin(x))^3-1/5*cos(x)/(a*(a+a*sin(x))^2)-1/5*cos(x)/(a^3+a^3*sin(x))],
[1/(a+a*sin(x))^3,x,3,-1/5*cos(x)/(a+a*sin(x))^3-2/15*cos(x)/(a*(a+a*sin(x))^2)-2/15*cos(x)/(a^3+a^3*sin(x))],
[csc(x)/(a+a*sin(x))^3,x,5,-arctanh(cos(x))/a^3+1/5*cos(x)/(a+a*sin(x))^3+7/15*cos(x)/(a*(a+a*sin(x))^2)+22/15*cos(x)/(a^3+a^3*sin(x))],
[csc(x)^2/(a+a*sin(x))^3,x,7,3*arctanh(cos(x))/a^3-24/5*cot(x)/a^3+1/5*cot(x)/(a+a*sin(x))^3+3/5*cot(x)/(a*(a+a*sin(x))^2)+3*cot(x)/(a^3+a^3*sin(x))],
[csc(x)^3/(a+a*sin(x))^3,x,8,-13/2*arctanh(cos(x))/a^3+152/15*cot(x)/a^3-13/2*cot(x)*csc(x)/a^3+1/5*cot(x)*csc(x)/(a+a*sin(x))^3+11/15*cot(x)*csc(x)/(a*(a+a*sin(x))^2)+76/15*cot(x)*csc(x)/(a^3+a^3*sin(x))],
[csc(x)^4/(a+a*sin(x))^3,x,8,23/2*arctanh(cos(x))/a^3-136/5*cot(x)/a^3-136/15*cot(x)^3/a^3+23/2*cot(x)*csc(x)/a^3+1/5*cot(x)*csc(x)^2/(a+a*sin(x))^3+13/15*cot(x)*csc(x)^2/(a*(a+a*sin(x))^2)+23/3*cot(x)*csc(x)^2/(a^3+a^3*sin(x))],

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

# n>0
[sin(c+d*x)^4*sqrt(a+a*sin(c+d*x)),x,5,-32/105*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/(a*d)-32/45*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-16/63*a*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-2/9*a*cos(c+d*x)*sin(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))+64/315*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sin(c+d*x)^3*sqrt(a+a*sin(c+d*x)),x,4,-12/35*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/(a*d)-4/5*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/7*a*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+8/35*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sin(c+d*x)^2*sqrt(a+a*sin(c+d*x)),x,3,-2/5*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/(a*d)-14/15*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+4/15*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sin(c+d*x)*sqrt(a+a*sin(c+d*x)),x,2,-2/3*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/3*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sqrt(a+a*sin(c+d*x)),x,1,-2*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)*sqrt(a+a*sin(c+d*x)),x,2,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d],
[csc(c+d*x)^2*sqrt(a+a*sin(c+d*x)),x,3,-arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)^3*sqrt(a+a*sin(c+d*x)),x,4,-3/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-3/4*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/2*a*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)^4*sqrt(a+a*sin(c+d*x)),x,5,-5/8*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-5/8*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-5/12*a*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/3*a*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)*sqrt(a-a*sin(c+d*x)),x,2,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a-a*sin(c+d*x)))*sqrt(a)/d],
[csc(c+d*x)*sqrt(-a+a*sin(c+d*x)),x,2,2*arctan(cos(c+d*x)*sqrt(a)/sqrt(-a+a*sin(c+d*x)))*sqrt(a)/d],
[csc(c+d*x)*sqrt(-a-a*sin(c+d*x)),x,2,2*arctan(cos(c+d*x)*sqrt(a)/sqrt(-a-a*sin(c+d*x)))*sqrt(a)/d],
[sin(c+d*x)^3*(a+a*sin(c+d*x))^(3/2),x,6,-68/105*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-68/45*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-34/63*a^2*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-2/9*a^2*cos(c+d*x)*sin(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))+136/315*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sin(c+d*x)^2*(a+a*sin(c+d*x))^(3/2),x,4,4/35*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-2/7*cos(c+d*x)*(a+a*sin(c+d*x))^(5/2)/(a*d)-152/105*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-38/105*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sin(c+d*x)*(a+a*sin(c+d*x))^(3/2),x,3,-2/5*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-8/5*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/5*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[(a+a*sin(c+d*x))^(3/2),x,2,-8/3*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/3*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[csc(c+d*x)*(a+a*sin(c+d*x))^(3/2),x,4,-2*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-2*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)^2*(a+a*sin(c+d*x))^(3/2),x,4,-3*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)^3*(a+a*sin(c+d*x))^(3/2),x,5,-7/4*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-7/4*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/2*a^2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)^4*(a+a*sin(c+d*x))^(3/2),x,6,-11/8*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-11/8*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-11/12*a^2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/3*a^2*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))],
[sin(c+d*x)^3*(a+a*sin(c+d*x))^(5/2),x,6,-284/231*a*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-284/99*a^3*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-710/693*a^3*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-46/99*a^3*cos(c+d*x)*sin(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))+568/693*a^2*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d-2/11*a^2*cos(c+d*x)*sin(c+d*x)^4*sqrt(a+a*sin(c+d*x))/d],
[sin(c+d*x)^2*(a+a*sin(c+d*x))^(5/2),x,5,-26/105*a*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d+4/63*cos(c+d*x)*(a+a*sin(c+d*x))^(5/2)/d-2/9*cos(c+d*x)*(a+a*sin(c+d*x))^(7/2)/(a*d)-832/315*a^3*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-208/315*a^2*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sin(c+d*x)*(a+a*sin(c+d*x))^(5/2),x,4,-2/7*a*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-2/7*cos(c+d*x)*(a+a*sin(c+d*x))^(5/2)/d-64/21*a^3*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-16/21*a^2*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[(a+a*sin(c+d*x))^(5/2),x,3,-2/5*a*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-64/15*a^3*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-16/15*a^2*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[csc(c+d*x)*(a+a*sin(c+d*x))^(5/2),x,4,-2*a^(5/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-14/3*a^3*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/3*a^2*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[csc(c+d*x)^2*(a+a*sin(c+d*x))^(5/2),x,4,-5*a^(5/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-a^3*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-a^2*cot(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[csc(c+d*x)^3*(a+a*sin(c+d*x))^(5/2),x,4,-19/4*a^(5/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-9/4*a^3*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/2*a^2*cot(c+d*x)*csc(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[csc(c+d*x)^4*(a+a*sin(c+d*x))^(5/2),x,5,-25/8*a^(5/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-25/8*a^3*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-13/12*a^3*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/3*a^2*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+a*sin(c+d*x))/d],
[csc(c+d*x)^5*(a+a*sin(c+d*x))^(5/2),x,6,-163/64*a^(5/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-163/64*a^3*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-163/96*a^3*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-17/24*a^3*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))-1/4*a^2*cot(c+d*x)*csc(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],

# n<0
[sin(c+d*x)^3/sqrt(a+a*sin(c+d*x)),x,6,arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(d*sqrt(a))-28/15*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/5*cos(c+d*x)*sin(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+2/15*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a*d)],
[sin(c+d*x)^2/sqrt(a+a*sin(c+d*x)),x,4,-arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(d*sqrt(a))+4/3*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/3*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a*d)],
[sin(c+d*x)/sqrt(a+a*sin(c+d*x)),x,3,arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(d*sqrt(a))-2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[1/sqrt(a+a*sin(c+d*x)),x,2,-arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[csc(c+d*x)/sqrt(a+a*sin(c+d*x)),x,5,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))+arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[csc(c+d*x)^2/sqrt(a+a*sin(c+d*x)),x,6,arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))-arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(d*sqrt(a))-cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)^3/sqrt(a+a*sin(c+d*x)),x,7,-7/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))+arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(d*sqrt(a))+1/4*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[sin(c+d*x)^4/(a+a*sin(c+d*x))^(3/2),x,7,1/2*cos(c+d*x)*sin(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))+15/2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))-31/5*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-9/10*cos(c+d*x)*sin(c+d*x)^2/(a*d*sqrt(a+a*sin(c+d*x)))+13/10*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[sin(c+d*x)^3/(a+a*sin(c+d*x))^(3/2),x,6,1/2*cos(c+d*x)*sin(c+d*x)^2/(d*(a+a*sin(c+d*x))^(3/2))-11/2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))+13/3*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-7/6*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[sin(c+d*x)^2/(a+a*sin(c+d*x))^(3/2),x,4,-1/2*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))+7/2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))-2*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))],
[sin(c+d*x)/(a+a*sin(c+d*x))^(3/2),x,3,1/2*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-3/2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[1/(a+a*sin(c+d*x))^(3/2),x,3,-1/2*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-1/2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[csc(c+d*x)/(a+a*sin(c+d*x))^(3/2),x,6,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)+1/2*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))+5/2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[csc(c+d*x)^2/(a+a*sin(c+d*x))^(3/2),x,7,3*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)+1/2*cot(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-9/2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))-3/2*cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)^3/(a+a*sin(c+d*x))^(3/2),x,8,-19/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)+1/2*cot(c+d*x)*csc(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))+13/2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))+7/4*cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-cot(c+d*x)*csc(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))],
[sin(c+d*x)^5/(a+a*sin(c+d*x))^(5/2),x,8,1/4*cos(c+d*x)*sin(c+d*x)^4/(d*(a+a*sin(c+d*x))^(5/2))+21/16*cos(c+d*x)*sin(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^(3/2))+283/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))-1729/120*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))-157/80*cos(c+d*x)*sin(c+d*x)^2/(a^2*d*sqrt(a+a*sin(c+d*x)))+787/240*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^3*d)],
[sin(c+d*x)^4/(a+a*sin(c+d*x))^(5/2),x,7,1/4*cos(c+d*x)*sin(c+d*x)^3/(d*(a+a*sin(c+d*x))^(5/2))+17/16*cos(c+d*x)*sin(c+d*x)^2/(a*d*(a+a*sin(c+d*x))^(3/2))-163/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))+197/24*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))-95/48*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^3*d)],
[sin(c+d*x)^3/(a+a*sin(c+d*x))^(5/2),x,6,1/4*cos(c+d*x)*sin(c+d*x)^2/(d*(a+a*sin(c+d*x))^(5/2))-13/16*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))+75/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))-9/4*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[sin(c+d*x)^2/(a+a*sin(c+d*x))^(5/2),x,4,-1/4*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(5/2))+13/16*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))-19/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[sin(c+d*x)/(a+a*sin(c+d*x))^(5/2),x,4,1/4*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(5/2))-5/16*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))-5/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[1/(a+a*sin(c+d*x))^(5/2),x,4,-1/4*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(5/2))-3/16*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))-3/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[csc(c+d*x)/(a+a*sin(c+d*x))^(5/2),x,7,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(5/2)*d)+1/4*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(5/2))+11/16*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))+43/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[csc(c+d*x)^2/(a+a*sin(c+d*x))^(5/2),x,8,5*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(5/2)*d)+1/4*cot(c+d*x)/(d*(a+a*sin(c+d*x))^(5/2))+15/16*cot(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))-115/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))-35/16*cot(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[csc(c+d*x)^3/(a+a*sin(c+d*x))^(5/2),x,9,-39/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(5/2)*d)+1/4*cot(c+d*x)*csc(c+d*x)/(d*(a+a*sin(c+d*x))^(5/2))+19/16*cot(c+d*x)*csc(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))+219/16*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))+63/16*cot(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))-31/16*cot(c+d*x)*csc(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],

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

# n>0
[sqrt(a+a*sin(e+f*x))/sqrt(sin(e+f*x)),x,2,-2*arcsin(cos(e+f*x)*sqrt(a)/sqrt(a+a*sin(e+f*x)))*sqrt(a)/f],
[sqrt(a-a*sin(e+f*x))/sqrt(-sin(e+f*x)),x,2,2*arcsin(cos(e+f*x)*sqrt(a)/sqrt(a-a*sin(e+f*x)))*sqrt(a)/f],

# n<0
[1/(sqrt(sin(x))*sqrt(1+sin(x))),x,2,-arcsin(cos(x)/(1+sin(x)))*sqrt(2)],
[1/(sqrt(sin(x))*sqrt(a+a*sin(x))),x,2,-arctan(cos(x)*sqrt(a)/(sqrt(2)*sqrt(sin(x))*sqrt(a+a*sin(x))))*sqrt(2)/sqrt(a)],
[1/(sqrt(1-sin(x))*sqrt(sin(x))),x,2,arctanh(cos(x)/(sqrt(2)*sqrt(1-sin(x))*sqrt(sin(x))))*sqrt(2)],
[1/(sqrt(sin(x))*sqrt(a-a*sin(x))),x,2,arctanh(cos(x)*sqrt(a)/(sqrt(2)*sqrt(sin(x))*sqrt(a-a*sin(x))))*sqrt(2)/sqrt(a)],

# Integrands of the form (d Sin[e+f x])^(n/3) (a+a Sin[e+f x])^m
[sin(c+d*x)^(1/3)/(a+a*sin(c+d*x))^2,x,5,-1/9*cos(c+d*x)*sin(c+d*x)^(1/3)/(a^2*d*(1+sin(c+d*x)))-1/3*cos(c+d*x)*sin(c+d*x)^(1/3)/(d*(a+a*sin(c+d*x))^2)+4/9*cos(c+d*x)*hypergeom([1/6,1/2],[7/6],sin(c+d*x)^2)*sin(c+d*x)^(1/3)/(a^2*d*sqrt(cos(c+d*x)^2))-1/36*cos(c+d*x)*hypergeom([1/2,2/3],[5/3],sin(c+d*x)^2)*sin(c+d*x)^(4/3)/(a^2*d*sqrt(cos(c+d*x)^2))],

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

# m>0
[sin(c+d*x)^3*(a+a*sin(c+d*x))^(2/3),x,6,-63/220*cos(c+d*x)*(a+a*sin(c+d*x))^(2/3)/d-3/11*cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x))^(2/3)/d-67/55*cos(c+d*x)*hypergeom([-1/6,1/2],[3/2],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(2/3)/(2^(5/6)*d*(1+sin(c+d*x))^(7/6))-3/44*cos(c+d*x)*(a+a*sin(c+d*x))^(5/3)/(a*d)],
[sin(c+d*x)^2*(a+a*sin(c+d*x))^(2/3),x,4,9/40*cos(c+d*x)*(a+a*sin(c+d*x))^(2/3)/d-19/10*cos(c+d*x)*hypergeom([-1/6,1/2],[3/2],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(2/3)/(2^(5/6)*d*(1+sin(c+d*x))^(7/6))-3/8*cos(c+d*x)*(a+a*sin(c+d*x))^(5/3)/(a*d)],
[sin(c+d*x)*(a+a*sin(c+d*x))^(2/3),x,3,-3/5*cos(c+d*x)*(a+a*sin(c+d*x))^(2/3)/d-4/5*2^(1/6)*cos(c+d*x)*hypergeom([-1/6,1/2],[3/2],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(2/3)/(d*(1+sin(c+d*x))^(7/6))],
[(a+a*sin(c+d*x))^(2/3),x,2,-2*2^(1/6)*cos(c+d*x)*hypergeom([-1/6,1/2],[3/2],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(2/3)/(d*(1+sin(c+d*x))^(7/6))],
[csc(c+d*x)*(a+a*sin(c+d*x))^(2/3),x,4,-2*2^(1/6)*AppellF1(1/2,1,-1/6,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)*(a+a*sin(c+d*x))^(2/3)/(d*(1+sin(c+d*x))^(7/6))],
[csc(c+d*x)^2*(a+a*sin(c+d*x))^(2/3),x,4,-2*2^(1/6)*AppellF1(1/2,2,-1/6,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)*(a+a*sin(c+d*x))^(2/3)/(d*(1+sin(c+d*x))^(7/6))],
[sin(c+d*x)^3*(a+a*sin(c+d*x))^(4/3),x,6,-388/455*2^(5/6)*a*cos(c+d*x)*hypergeom([-5/6,1/2],[3/2],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(1/3)/(d*(1+sin(c+d*x))^(5/6))-72/455*cos(c+d*x)*(a+a*sin(c+d*x))^(4/3)/d-3/13*cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x))^(4/3)/d-6/65*cos(c+d*x)*(a+a*sin(c+d*x))^(7/3)/(a*d)],
[sin(c+d*x)^2*(a+a*sin(c+d*x))^(4/3),x,4,-37/35*2^(5/6)*a*cos(c+d*x)*hypergeom([-5/6,1/2],[3/2],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(1/3)/(d*(1+sin(c+d*x))^(5/6))+9/70*cos(c+d*x)*(a+a*sin(c+d*x))^(4/3)/d-3/10*cos(c+d*x)*(a+a*sin(c+d*x))^(7/3)/(a*d)],
[sin(c+d*x)*(a+a*sin(c+d*x))^(4/3),x,3,-8/7*2^(5/6)*a*cos(c+d*x)*hypergeom([-5/6,1/2],[3/2],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(1/3)/(d*(1+sin(c+d*x))^(5/6))-3/7*cos(c+d*x)*(a+a*sin(c+d*x))^(4/3)/d],
[(a+a*sin(c+d*x))^(4/3),x,2,-2*2^(5/6)*a*cos(c+d*x)*hypergeom([-5/6,1/2],[3/2],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(1/3)/(d*(1+sin(c+d*x))^(5/6))],
[csc(c+d*x)*(a+a*sin(c+d*x))^(4/3),x,4,-2*2^(5/6)*a*AppellF1(1/2,1,-5/6,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)*(a+a*sin(c+d*x))^(1/3)/(d*(1+sin(c+d*x))^(5/6))],
[csc(c+d*x)^2*(a+a*sin(c+d*x))^(4/3),x,4,-2*2^(5/6)*a*AppellF1(1/2,2,-5/6,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)*(a+a*sin(c+d*x))^(1/3)/(d*(1+sin(c+d*x))^(5/6))],

# m<0
[sin(c+d*x)^3/(a+a*sin(c+d*x))^(1/3),x,6,-99/80*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(1/3))-3/8*cos(c+d*x)*sin(c+d*x)^2/(d*(a+a*sin(c+d*x))^(1/3))+37/40*cos(c+d*x)*hypergeom([1/2,5/6],[3/2],1/2*(1-sin(c+d*x)))/(2^(5/6)*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))+3/40*cos(c+d*x)*(a+a*sin(c+d*x))^(2/3)/(a*d)],
[sin(c+d*x)^2/(a+a*sin(c+d*x))^(1/3),x,4,9/10*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(1/3))-7/5*cos(c+d*x)*hypergeom([1/2,5/6],[3/2],1/2*(1-sin(c+d*x)))/(2^(5/6)*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))-3/5*cos(c+d*x)*(a+a*sin(c+d*x))^(2/3)/(a*d)],
[sin(c+d*x)/(a+a*sin(c+d*x))^(1/3),x,3,-3/2*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(1/3))+cos(c+d*x)*hypergeom([1/2,5/6],[3/2],1/2*(1-sin(c+d*x)))/(2^(5/6)*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[1/(a+a*sin(c+d*x))^(1/3),x,2,-2^(1/6)*cos(c+d*x)*hypergeom([1/2,5/6],[3/2],1/2*(1-sin(c+d*x)))/(d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[csc(c+d*x)/(a+a*sin(c+d*x))^(1/3),x,4,-2^(1/6)*AppellF1(1/2,1,5/6,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)/(d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[csc(c+d*x)^2/(a+a*sin(c+d*x))^(1/3),x,4,-2^(1/6)*AppellF1(1/2,2,5/6,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)/(d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[sin(c+d*x)^3/(a+a*sin(c+d*x))^(4/3),x,6,6/5*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(4/3))-3/5*cos(c+d*x)*sin(c+d*x)^2/(d*(a+a*sin(c+d*x))^(4/3))+6/5*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(1/3))-2*2^(1/6)*cos(c+d*x)*hypergeom([1/2,5/6],[3/2],1/2*(1-sin(c+d*x)))/(a*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[sin(c+d*x)^2/(a+a*sin(c+d*x))^(4/3),x,4,-3/5*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(4/3))-3/2*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(1/3))+13/5*cos(c+d*x)*hypergeom([1/2,5/6],[3/2],1/2*(1-sin(c+d*x)))/(2^(5/6)*a*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[sin(c+d*x)/(a+a*sin(c+d*x))^(4/3),x,3,3/5*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(4/3))-4/5*2^(1/6)*cos(c+d*x)*hypergeom([1/2,5/6],[3/2],1/2*(1-sin(c+d*x)))/(a*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[1/(a+a*sin(c+d*x))^(4/3),x,2,-cos(c+d*x)*hypergeom([1/2,11/6],[3/2],1/2*(1-sin(c+d*x)))/(2^(5/6)*a*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[csc(c+d*x)/(a+a*sin(c+d*x))^(4/3),x,4,-AppellF1(1/2,1,11/6,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)/(2^(5/6)*a*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],
[csc(c+d*x)^2/(a+a*sin(c+d*x))^(4/3),x,4,-AppellF1(1/2,2,11/6,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)/(2^(5/6)*a*d*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(1/3))],

# Integrands of the form (d Sin[e+f x])^n (a+a Sin[e+f x])^m with n symbolic
[sin(e+f*x)^n*(1+sin(e+f*x))^(3/2),x,4,-2*(5+4*n)*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))/(f*(3+2*n)*sqrt(1+sin(e+f*x)))-2*cos(e+f*x)*sin(e+f*x)^(1+n)/(f*(3+2*n)*sqrt(1+sin(e+f*x)))],
[sin(e+f*x)^n*(1+sin(e+f*x))^(1/2),x,2,-2*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))/(f*sqrt(1+sin(e+f*x)))],
[sin(e+f*x)^n/(1+sin(e+f*x))^(1/2),x,3,-AppellF1(1/2,-n,1,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)/(f*sqrt(1+sin(e+f*x)))],
[sin(e+f*x)^n/(1+sin(e+f*x))^(3/2),x,3,-1/2*AppellF1(1/2,-n,2,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)/(f*sqrt(1+sin(e+f*x)))],
[sin(e+f*x)^n*(a+a*sin(e+f*x))^(3/2),x,4,-2*a^2*(5+4*n)*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))/(f*(3+2*n)*sqrt(a+a*sin(e+f*x)))-2*a^2*cos(e+f*x)*sin(e+f*x)^(1+n)/(f*(3+2*n)*sqrt(a+a*sin(e+f*x)))],
[sin(e+f*x)^n*(a+a*sin(e+f*x))^(1/2),x,2,-2*a*cos(e+f*x)*hypergeom([1/2,-n],[3/2],1-sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[sin(e+f*x)^n/(a+a*sin(e+f*x))^(1/2),x,4,-AppellF1(1/2,-n,1,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))],
[sin(e+f*x)^n/(a+a*sin(e+f*x))^(3/2),x,4,-1/2*AppellF1(1/2,-n,2,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)/(a*f*sqrt(a+a*sin(e+f*x)))],
[(d*sin(e+f*x))^n*(1+sin(e+f*x))^(3/2),x,4,-2*cos(e+f*x)*(d*sin(e+f*x))^(1+n)/(d*f*(3+2*n)*sqrt(1+sin(e+f*x)))+(5+4*n)*cos(e+f*x)*hypergeom([1/2,1+n],[2+n],sin(e+f*x))*(d*sin(e+f*x))^(1+n)/(d*f*(1+n)*(3+2*n)*sqrt(1-sin(e+f*x))*sqrt(1+sin(e+f*x)))],
[(d*sin(e+f*x))^n*(1+sin(e+f*x))^(1/2),x,2,cos(e+f*x)*hypergeom([1/2,1+n],[2+n],sin(e+f*x))*(d*sin(e+f*x))^(1+n)/(d*f*(1+n)*sqrt(1-sin(e+f*x))*sqrt(1+sin(e+f*x)))],
[(d*sin(e+f*x))^n/(1+sin(e+f*x))^(1/2),x,4,-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(1+sin(e+f*x)))],
[(d*sin(e+f*x))^n/(1+sin(e+f*x))^(3/2),x,4,-1/2*AppellF1(1/2,-n,2,3/2,1-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(1+sin(e+f*x)))],
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^(3/2),x,5,-2*a^2*(5+4*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^2*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+a*sin(e+f*x))^(1/2),x,3,-2*a*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+a*sin(e+f*x))^(1/2),x,5,-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)))],
[(d*sin(e+f*x))^n/(a+a*sin(e+f*x))^(3/2),x,5,-1/2*AppellF1(1/2,-n,2,3/2,1-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)))],

# Integrands of the form Sin[e+f x]^n (a+a Sin[e+f x])^m with m symbolic
[sin(e+f*x)^n*(1+sin(e+f*x))^m,x,2,-2^(1/2+m)*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)/(f*sqrt(1+sin(e+f*x)))],
[(1-sin(e+f*x))^m*(-sin(e+f*x))^n,x,2,2^(1/2+m)*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)/(f*sqrt(1-sin(e+f*x)))],
[(d*sin(e+f*x))^n*(1+sin(e+f*x))^m,x,3,-2^(1/2+m)*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/(f*sin(e+f*x)^n*sqrt(1+sin(e+f*x)))],
[(1-sin(e+f*x))^m*(d*sin(e+f*x))^n,x,3,2^(1/2+m)*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/(f*(-sin(e+f*x))^n*sqrt(1-sin(e+f*x)))],
[sin(e+f*x)^n*(a+a*sin(e+f*x))^m,x,3,-2^(1/2+m)*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)*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/f],
[(-sin(e+f*x))^n*(a-a*sin(e+f*x))^m,x,3,2^(1/2+m)*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)*(1-sin(e+f*x))^(-1/2-m)*(a-a*sin(e+f*x))^m/f],
[(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m,x,4,-2^(1/2+m)*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))^m,x,4,2^(1/2+m)*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)*(1-sin(e+f*x))^(-1/2-m)*(d*sin(e+f*x))^n*(a-a*sin(e+f*x))^m/(f*(-sin(e+f*x))^n)],
[sin(c+d*x)^4*(a+a*sin(c+d*x))^n,x,7,(9-n-n^2)*cos(c+d*x)*(a+a*sin(c+d*x))^n/(d*(24+50*n+35*n^2+10*n^3+n^4))-n*cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x))^n/(d*(3+n)*(4+n))-cos(c+d*x)*sin(c+d*x)^3*(a+a*sin(c+d*x))^n/(d*(4+n))-2^(1/2+n)*(9+12*n+17*n^2+6*n^3+n^4)*cos(c+d*x)*hypergeom([1/2,1/2-n],[3/2],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-1/2-n)*(a+a*sin(c+d*x))^n/(d*(4+n)*(6+11*n+6*n^2+n^3))-(9+3*n+n^2)*cos(c+d*x)*(a+a*sin(c+d*x))^(1+n)/(a*d*(4+n)*(6+5*n+n^2))],
[sin(c+d*x)^3*(a+a*sin(c+d*x))^n,x,6,-(4+n)*cos(c+d*x)*(a+a*sin(c+d*x))^n/(d*(6+11*n+6*n^2+n^3))-cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x))^n/(d*(3+n))-2^(1/2+n)*n*(5+3*n+n^2)*cos(c+d*x)*hypergeom([1/2,1/2-n],[3/2],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-1/2-n)*(a+a*sin(c+d*x))^n/(d*(6+11*n+6*n^2+n^3))-n*cos(c+d*x)*(a+a*sin(c+d*x))^(1+n)/(a*d*(6+5*n+n^2))],
[sin(c+d*x)^2*(a+a*sin(c+d*x))^n,x,4,cos(c+d*x)*(a+a*sin(c+d*x))^n/(d*(2+3*n+n^2))-2^(1/2+n)*(1+n+n^2)*cos(c+d*x)*hypergeom([1/2,1/2-n],[3/2],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-1/2-n)*(a+a*sin(c+d*x))^n/(d*(2+3*n+n^2))-cos(c+d*x)*(a+a*sin(c+d*x))^(1+n)/(a*d*(2+n))],
[sin(c+d*x)*(a+a*sin(c+d*x))^n,x,3,-cos(c+d*x)*(a+a*sin(c+d*x))^n/(d*(1+n))-2^(1/2+n)*n*cos(c+d*x)*hypergeom([1/2,1/2-n],[3/2],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-1/2-n)*(a+a*sin(c+d*x))^n/(d*(1+n))],
[(a+a*sin(c+d*x))^n,x,2,-2^(1/2+n)*cos(c+d*x)*hypergeom([1/2,1/2-n],[3/2],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-1/2-n)*(a+a*sin(c+d*x))^n/d],
[csc(c+d*x)*(a+a*sin(c+d*x))^n,x,4,-2^(1/2+n)*AppellF1(1/2,1,1/2-n,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)*(1+sin(c+d*x))^(-1/2-n)*(a+a*sin(c+d*x))^n/d],
[csc(c+d*x)^2*(a+a*sin(c+d*x))^n,x,4,-2^(1/2+n)*AppellF1(1/2,2,1/2-n,3/2,1-sin(c+d*x),1/2*(1-sin(c+d*x)))*cos(c+d*x)*(1+sin(c+d*x))^(-1/2-n)*(a+a*sin(c+d*x))^n/d],
[(1+sin(c+d*x))^n,x,1,-2^(1/2+n)*cos(c+d*x)*hypergeom([1/2,1/2-n],[3/2],1/2*(1-sin(c+d*x)))/(d*sqrt(1+sin(c+d*x)))],
[(1-sin(c+d*x))^n,x,1,2^(1/2+n)*cos(c+d*x)*hypergeom([1/2,1/2-n],[3/2],1/2*(1+sin(c+d*x)))/(d*sqrt(1-sin(c+d*x)))],

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

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

# n>0
[sin(e+f*x)^3*(a+b*sin(e+f*x)),x,6,3/8*b*x-a*cos(e+f*x)/f+1/3*a*cos(e+f*x)^3/f-3/8*b*cos(e+f*x)*sin(e+f*x)/f-1/4*b*cos(e+f*x)*sin(e+f*x)^3/f],
[sin(e+f*x)^2*(a+b*sin(e+f*x)),x,5,1/2*a*x-b*cos(e+f*x)/f+1/3*b*cos(e+f*x)^3/f-1/2*a*cos(e+f*x)*sin(e+f*x)/f],
[sin(e+f*x)*(a+b*sin(e+f*x)),x,1,1/2*b*x-a*cos(e+f*x)/f-1/2*b*cos(e+f*x)*sin(e+f*x)/f],
[a+b*sin(e+f*x),x,2,a*x-b*cos(e+f*x)/f],
[csc(e+f*x)*(a+b*sin(e+f*x)),x,2,b*x-a*arctanh(cos(e+f*x))/f],
[csc(e+f*x)^2*(a+b*sin(e+f*x)),x,4,-b*arctanh(cos(e+f*x))/f-a*cot(e+f*x)/f],
[csc(e+f*x)^3*(a+b*sin(e+f*x)),x,5,-1/2*a*arctanh(cos(e+f*x))/f-b*cot(e+f*x)/f-1/2*a*cot(e+f*x)*csc(e+f*x)/f],
[csc(e+f*x)^4*(a+b*sin(e+f*x)),x,5,-1/2*b*arctanh(cos(e+f*x))/f-a*cot(e+f*x)/f-1/3*a*cot(e+f*x)^3/f-1/2*b*cot(e+f*x)*csc(e+f*x)/f],
[sin(e+f*x)^3*(a+b*sin(e+f*x))^2,x,7,3/4*a*b*x-(a^2+b^2)*cos(e+f*x)/f+1/3*(a^2+2*b^2)*cos(e+f*x)^3/f-1/5*b^2*cos(e+f*x)^5/f-3/4*a*b*cos(e+f*x)*sin(e+f*x)/f-1/2*a*b*cos(e+f*x)*sin(e+f*x)^3/f],
[sin(e+f*x)^2*(a+b*sin(e+f*x))^2,x,6,1/8*(4*a^2+3*b^2)*x-2*a*b*cos(e+f*x)/f+2/3*a*b*cos(e+f*x)^3/f-1/8*(4*a^2+3*b^2)*cos(e+f*x)*sin(e+f*x)/f-1/4*b^2*cos(e+f*x)*sin(e+f*x)^3/f],
[sin(e+f*x)*(a+b*sin(e+f*x))^2,x,2,a*b*x-2/3*(a^2+b^2)*cos(e+f*x)/f-1/3*a*b*cos(e+f*x)*sin(e+f*x)/f-1/3*cos(e+f*x)*(a+b*sin(e+f*x))^2/f],
[(a+b*sin(e+f*x))^2,x,1,1/2*(2*a^2+b^2)*x-2*a*b*cos(e+f*x)/f-1/2*b^2*cos(e+f*x)*sin(e+f*x)/f],
[csc(e+f*x)*(a+b*sin(e+f*x))^2,x,3,2*a*b*x-a^2*arctanh(cos(e+f*x))/f-b^2*cos(e+f*x)/f],
[csc(e+f*x)^2*(a+b*sin(e+f*x))^2,x,4,b^2*x-2*a*b*arctanh(cos(e+f*x))/f-a^2*cot(e+f*x)/f],
[csc(e+f*x)^3*(a+b*sin(e+f*x))^2,x,5,-1/2*(a^2+2*b^2)*arctanh(cos(e+f*x))/f-2*a*b*cot(e+f*x)/f-1/2*a^2*cot(e+f*x)*csc(e+f*x)/f],
[csc(e+f*x)^4*(a+b*sin(e+f*x))^2,x,6,-a*b*arctanh(cos(e+f*x))/f-1/3*(2*a^2+3*b^2)*cot(e+f*x)/f-a*b*cot(e+f*x)*csc(e+f*x)/f-1/3*a^2*cot(e+f*x)*csc(e+f*x)^2/f],
[csc(e+f*x)^5*(a+b*sin(e+f*x))^2,x,6,-1/8*(3*a^2+4*b^2)*arctanh(cos(e+f*x))/f-2*a*b*cot(e+f*x)/f-2/3*a*b*cot(e+f*x)^3/f-1/8*(3*a^2+4*b^2)*cot(e+f*x)*csc(e+f*x)/f-1/4*a^2*cot(e+f*x)*csc(e+f*x)^3/f],
[sin(e+f*x)^3*(a+b*sin(e+f*x))^3,x,8,1/16*b*(18*a^2+5*b^2)*x-a*(a^2+3*b^2)*cos(e+f*x)/f+1/3*a*(a^2+6*b^2)*cos(e+f*x)^3/f-3/5*a*b^2*cos(e+f*x)^5/f-1/16*b*(18*a^2+5*b^2)*cos(e+f*x)*sin(e+f*x)/f-1/24*b*(18*a^2+5*b^2)*cos(e+f*x)*sin(e+f*x)^3/f-1/6*b^3*cos(e+f*x)*sin(e+f*x)^5/f,1/16*b*(18*a^2+5*b^2)*x-1/5*a*(5*a^2+12*b^2)*cos(e+f*x)/f+1/15*a*(5*a^2+12*b^2)*cos(e+f*x)^3/f-1/16*b*(18*a^2+5*b^2)*cos(e+f*x)*sin(e+f*x)/f-1/24*b*(18*a^2+5*b^2)*cos(e+f*x)*sin(e+f*x)^3/f-13/30*a*b^2*cos(e+f*x)*sin(e+f*x)^4/f-1/6*b^2*cos(e+f*x)*sin(e+f*x)^4*(a+b*sin(e+f*x))/f],
[sin(e+f*x)^2*(a+b*sin(e+f*x))^3,x,4,1/8*a*(4*a^2+9*b^2)*x-1/5*b*(15*a^2+4*b^2)*cos(e+f*x)/f+1/15*b*(15*a^2+4*b^2)*cos(e+f*x)^3/f-1/8*a*(4*a^2+9*b^2)*cos(e+f*x)*sin(e+f*x)/f-11/20*a*b^2*cos(e+f*x)*sin(e+f*x)^3/f-1/5*b^2*cos(e+f*x)*sin(e+f*x)^3*(a+b*sin(e+f*x))/f,1/8*a*(4*a^2+9*b^2)*x+1/30*(3*a^4-52*a^2*b^2-16*b^4)*cos(e+f*x)/(b*f)+1/120*a*(6*a^2-71*b^2)*cos(e+f*x)*sin(e+f*x)/f+1/60*(3*a^2-16*b^2)*cos(e+f*x)*(a+b*sin(e+f*x))^2/(b*f)+1/20*a*cos(e+f*x)*(a+b*sin(e+f*x))^3/(b*f)-1/5*cos(e+f*x)*(a+b*sin(e+f*x))^4/(b*f)],
[sin(e+f*x)*(a+b*sin(e+f*x))^3,x,3,3/8*b*(4*a^2+b^2)*x-1/2*a*(a^2+4*b^2)*cos(e+f*x)/f-1/8*b*(2*a^2+3*b^2)*cos(e+f*x)*sin(e+f*x)/f-1/4*a*cos(e+f*x)*(a+b*sin(e+f*x))^2/f-1/4*cos(e+f*x)*(a+b*sin(e+f*x))^3/f],
[(a+b*sin(e+f*x))^3,x,2,1/2*a*(2*a^2+3*b^2)*x-2/3*b*(4*a^2+b^2)*cos(e+f*x)/f-5/6*a*b^2*cos(e+f*x)*sin(e+f*x)/f-1/3*b*cos(e+f*x)*(a+b*sin(e+f*x))^2/f],
[csc(e+f*x)*(a+b*sin(e+f*x))^3,x,4,1/2*b*(6*a^2+b^2)*x-a^3*arctanh(cos(e+f*x))/f-5/2*a*b^2*cos(e+f*x)/f-1/2*b^2*cos(e+f*x)*(a+b*sin(e+f*x))/f],
[csc(e+f*x)^2*(a+b*sin(e+f*x))^3,x,4,3*a*b^2*x-3*a^2*b*arctanh(cos(e+f*x))/f+b*(a^2-b^2)*cos(e+f*x)/f-a^2*cot(e+f*x)*(a+b*sin(e+f*x))/f],
[csc(e+f*x)^3*(a+b*sin(e+f*x))^3,x,4,b^3*x-1/2*a*(a^2+6*b^2)*arctanh(cos(e+f*x))/f-5/2*a^2*b*cot(e+f*x)/f-1/2*a^2*cot(e+f*x)*csc(e+f*x)*(a+b*sin(e+f*x))/f],
[csc(e+f*x)^4*(a+b*sin(e+f*x))^3,x,6,-1/2*b*(3*a^2+2*b^2)*arctanh(cos(e+f*x))/f-1/3*a*(2*a^2+9*b^2)*cot(e+f*x)/f-7/6*a^2*b*cot(e+f*x)*csc(e+f*x)/f-1/3*a^2*cot(e+f*x)*csc(e+f*x)^2*(a+b*sin(e+f*x))/f],
[csc(e+f*x)^5*(a+b*sin(e+f*x))^3,x,7,-3/8*a*(a^2+4*b^2)*arctanh(cos(e+f*x))/f-b*(2*a^2+b^2)*cot(e+f*x)/f-3/8*a*(a^2+4*b^2)*cot(e+f*x)*csc(e+f*x)/f-3/4*a^2*b*cot(e+f*x)*csc(e+f*x)^2/f-1/4*a^2*cot(e+f*x)*csc(e+f*x)^3*(a+b*sin(e+f*x))/f],
[(a+b*sin(e+f*x))^4,x,3,1/8*(8*a^4+24*a^2*b^2+3*b^4)*x-1/6*a*b*(19*a^2+16*b^2)*cos(e+f*x)/f-1/24*b^2*(26*a^2+9*b^2)*cos(e+f*x)*sin(e+f*x)/f-7/12*a*b*cos(e+f*x)*(a+b*sin(e+f*x))^2/f-1/4*b*cos(e+f*x)*(a+b*sin(e+f*x))^3/f],

# n<0
[sin(x)^4/(a+b*sin(x)),x,7,-1/2*a*(2*a^2+b^2)*x/b^4-1/3*(3*a^2+2*b^2)*cos(x)/b^3+1/2*a*cos(x)*sin(x)/b^2-1/3*cos(x)*sin(x)^2/b+2*a^4*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^4*sqrt(a^2-b^2))],
[sin(x)^3/(a+b*sin(x)),x,6,1/2*(2*a^2+b^2)*x/b^3+a*cos(x)/b^2-1/2*cos(x)*sin(x)/b-2*a^3*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^3*sqrt(a^2-b^2))],
[sin(x)^2/(a+b*sin(x)),x,6,-a*x/b^2-cos(x)/b+2*a^2*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^2*sqrt(a^2-b^2))],
[sin(x)/(a+b*sin(x)),x,4,x/b-2*a*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b*sqrt(a^2-b^2))],
[1/(a+b*sin(x)),x,3,2*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/sqrt(a^2-b^2)],
[csc(x)/(a+b*sin(x)),x,5,-arctanh(cos(x))/a-2*b*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a*sqrt(a^2-b^2))],
[csc(x)^2/(a+b*sin(x)),x,7,b*arctanh(cos(x))/a^2-cot(x)/a+2*b^2*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^2*sqrt(a^2-b^2))],
[csc(x)^3/(a+b*sin(x)),x,7,-1/2*(a^2+2*b^2)*arctanh(cos(x))/a^3+b*cot(x)/a^2-1/2*cot(x)*csc(x)/a-2*b^3*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^3*sqrt(a^2-b^2))],
[csc(x)^4/(a+b*sin(x)),x,8,1/2*b*(a^2+2*b^2)*arctanh(cos(x))/a^4-1/3*(2*a^2+3*b^2)*cot(x)/a^3+1/2*b*cot(x)*csc(x)/a^2-1/3*cot(x)*csc(x)^2/a+2*b^4*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^4*sqrt(a^2-b^2))],
[sin(x)^4/(a+b*sin(x))^2,x,7,1/2*(6*a^2+b^2)*x/b^4-2*a^3*(3*a^2-4*b^2)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^4*(a^2-b^2)^(3/2))+a*(3*a^2-2*b^2)*cos(x)/(b^3*(a^2-b^2))-1/2*(3*a^2-b^2)*cos(x)*sin(x)/(b^2*(a^2-b^2))+a^2*cos(x)*sin(x)^2/(b*(a^2-b^2)*(a+b*sin(x)))],
[sin(x)^3/(a+b*sin(x))^2,x,6,-2*a*x/b^3+2*a^2*(2*a^2-3*b^2)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^3*(a^2-b^2)^(3/2))-(2*a^2-b^2)*cos(x)/(b^2*(a^2-b^2))+a^2*cos(x)*sin(x)/(b*(a^2-b^2)*(a+b*sin(x)))],
[sin(x)^2/(a+b*sin(x))^2,x,5,x/b^2-2*a*(a^2-2*b^2)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^2*(a^2-b^2)^(3/2))+a^2*cos(x)/(b*(a^2-b^2)*(a+b*sin(x)))],
[sin(x)/(a+b*sin(x))^2,x,5,-2*b*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^2-b^2)^(3/2)-a*cos(x)/((a^2-b^2)*(a+b*sin(x)))],
[1/(a+b*sin(x))^2,x,5,2*a*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^2-b^2)^(3/2)+b*cos(x)/((a^2-b^2)*(a+b*sin(x)))],
[csc(x)/(a+b*sin(x))^2,x,6,-2*b*(2*a^2-b^2)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(3/2))-arctanh(cos(x))/a^2-b^2*cos(x)/(a*(a^2-b^2)*(a+b*sin(x)))],
[csc(x)^2/(a+b*sin(x))^2,x,7,2*b^2*(3*a^2-2*b^2)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(3/2))+2*b*arctanh(cos(x))/a^3-(a^2-2*b^2)*cot(x)/(a^2*(a^2-b^2))-b^2*cot(x)/(a*(a^2-b^2)*(a+b*sin(x)))],
[csc(x)^3/(a+b*sin(x))^2,x,8,-2*b^3*(4*a^2-3*b^2)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^4*(a^2-b^2)^(3/2))-1/2*(a^2+6*b^2)*arctanh(cos(x))/a^4+b*(2*a^2-3*b^2)*cot(x)/(a^3*(a^2-b^2))-1/2*(a^2-3*b^2)*cot(x)*csc(x)/(a^2*(a^2-b^2))-b^2*cot(x)*csc(x)/(a*(a^2-b^2)*(a+b*sin(x)))],
[sin(x)^5/(a+b*sin(x))^3,x,8,1/2*(12*a^2+b^2)*x/b^5-a^3*(12*a^4-29*a^2*b^2+20*b^4)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^5*(a^2-b^2)^(5/2))+3/2*a*(4*a^4-7*a^2*b^2+2*b^4)*cos(x)/(b^4*(a^2-b^2)^2)-1/2*(6*a^4-10*a^2*b^2+b^4)*cos(x)*sin(x)/(b^3*(a^2-b^2)^2)+1/2*a^2*cos(x)*sin(x)^3/(b*(a^2-b^2)*(a+b*sin(x))^2)+1/2*a^2*(4*a^2-7*b^2)*cos(x)*sin(x)^2/(b^2*(a^2-b^2)^2*(a+b*sin(x)))],
[sin(x)^4/(a+b*sin(x))^3,x,7,-3*a*x/b^4+3*a^2*(2*a^4-5*a^2*b^2+4*b^4)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^4*(a^2-b^2)^(5/2))-1/2*(3*a^2-2*b^2)*cos(x)/(b^3*(a^2-b^2))+1/2*a^2*cos(x)*sin(x)^2/(b*(a^2-b^2)*(a+b*sin(x))^2)-3/2*a^3*(a^2-2*b^2)*cos(x)/(b^3*(a^2-b^2)^2*(a+b*sin(x)))],
[sin(x)^3/(a+b*sin(x))^3,x,6,x/b^3-a*(2*a^4-5*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(b^3*(a^2-b^2)^(5/2))+1/2*a^2*cos(x)*sin(x)/(b*(a^2-b^2)*(a+b*sin(x))^2)+1/2*a^2*(2*a^2-5*b^2)*cos(x)/(b^2*(a^2-b^2)^2*(a+b*sin(x)))],
[sin(x)^2/(a+b*sin(x))^3,x,6,(a^2+2*b^2)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^2-b^2)^(5/2)+1/2*a^2*cos(x)/(b*(a^2-b^2)*(a+b*sin(x))^2)-1/2*a*(a^2-4*b^2)*cos(x)/(b*(a^2-b^2)^2*(a+b*sin(x)))],
[sin(x)/(a+b*sin(x))^3,x,6,-3*a*b*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^2-b^2)^(5/2)-1/2*a*cos(x)/((a^2-b^2)*(a+b*sin(x))^2)-1/2*(a^2+2*b^2)*cos(x)/((a^2-b^2)^2*(a+b*sin(x)))],
[1/(a+b*sin(x))^3,x,6,(2*a^2+b^2)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^2-b^2)^(5/2)+1/2*b*cos(x)/((a^2-b^2)*(a+b*sin(x))^2)+3/2*a*b*cos(x)/((a^2-b^2)^2*(a+b*sin(x)))],
[csc(x)/(a+b*sin(x))^3,x,7,-b*(6*a^4-5*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(5/2))-arctanh(cos(x))/a^3-1/2*b^2*cos(x)/(a*(a^2-b^2)*(a+b*sin(x))^2)-1/2*b^2*(5*a^2-2*b^2)*cos(x)/(a^2*(a^2-b^2)^2*(a+b*sin(x)))],
[csc(x)^2/(a+b*sin(x))^3,x,8,3*b^2*(4*a^4-5*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^4*(a^2-b^2)^(5/2))+3*b*arctanh(cos(x))/a^4-1/2*(2*a^4-11*a^2*b^2+6*b^4)*cot(x)/(a^3*(a^2-b^2)^2)-1/2*b^2*cot(x)/(a*(a^2-b^2)*(a+b*sin(x))^2)-3/2*b^2*(2*a^2-b^2)*cot(x)/(a^2*(a^2-b^2)^2*(a+b*sin(x)))],
[csc(x)^3/(a+b*sin(x))^3,x,9,-b^3*(20*a^4-29*a^2*b^2+12*b^4)*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))/(a^5*(a^2-b^2)^(5/2))-1/2*(a^2+12*b^2)*arctanh(cos(x))/a^5+3/2*b*(2*a^4-7*a^2*b^2+4*b^4)*cot(x)/(a^4*(a^2-b^2)^2)-1/2*(a^4-10*a^2*b^2+6*b^4)*cot(x)*csc(x)/(a^3*(a^2-b^2)^2)-1/2*b^2*cot(x)*csc(x)/(a*(a^2-b^2)*(a+b*sin(x))^2)-1/2*b^2*(7*a^2-4*b^2)*cot(x)*csc(x)/(a^2*(a^2-b^2)^2*(a+b*sin(x)))],
[1/(a+b*sin(c+d*x))^4,x,7,a*(2*a^2+3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)+1/3*b*cos(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x))^3)+5/6*a*b*cos(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)+1/6*b*(11*a^2+4*b^2)*cos(c+d*x)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))],

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

# n>0
[sin(e+f*x)*sqrt(a+b*sin(e+f*x)),x,6,-2/3*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/f+2/3*a*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(e+f*x))/(b*f*sqrt((a+b*sin(e+f*x))/(a+b)))-2/3*(a^2-b^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(b*f*sqrt(a+b*sin(e+f*x)))],
[sqrt(a+b*sin(e+f*x)),x,2,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(e+f*x))/(f*sqrt((a+b*sin(e+f*x))/(a+b)))],
[csc(e+f*x)*sqrt(a+b*sin(e+f*x)),x,5,2*b*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(f*sqrt(a+b*sin(e+f*x)))+2*a*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(f*sqrt(a+b*sin(e+f*x)))],
[csc(e+f*x)^2*sqrt(a+b*sin(e+f*x)),x,9,-cot(e+f*x)*sqrt(a+b*sin(e+f*x))/f-sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(e+f*x))/(f*sqrt((a+b*sin(e+f*x))/(a+b)))+a*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(f*sqrt(a+b*sin(e+f*x)))+b*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(f*sqrt(a+b*sin(e+f*x)))],

# n<0
[sin(e+f*x)/sqrt(a+b*sin(e+f*x)),x,5,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(e+f*x))/(b*f*sqrt((a+b*sin(e+f*x))/(a+b)))-2*a*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(b*f*sqrt(a+b*sin(e+f*x)))],
[1/sqrt(a+b*sin(e+f*x)),x,2,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(f*sqrt(a+b*sin(e+f*x)))],
[csc(e+f*x)/sqrt(a+b*sin(e+f*x)),x,2,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(f*sqrt(a+b*sin(e+f*x)))],
[csc(e+f*x)^2/sqrt(a+b*sin(e+f*x)),x,9,-cot(e+f*x)*sqrt(a+b*sin(e+f*x))/(a*f)-sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(e+f*x))/(a*f*sqrt((a+b*sin(e+f*x))/(a+b)))+sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(f*sqrt(a+b*sin(e+f*x)))-b*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(e+f*x))/(a+b))/(a*f*sqrt(a+b*sin(e+f*x)))],

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

# Integrands of the form (d Sin[e+f x])^(m/2) (a+b Sin[e+f x])^(n/2)
[sqrt(sin(c+d*x))*sqrt(a+b*sin(c+d*x)),x,7,-cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(d*sqrt(sin(c+d*x)))+(a-b)*EllipticE(sqrt(a+b*sin(c+d*x))/(sqrt(a+b)*sqrt(sin(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-csc(c+d*x))/(a+b))*sqrt(a*(1+csc(c+d*x))/(a-b))*tan(c+d*x)/(a*d)-EllipticF(sqrt(a+b*sin(c+d*x))/(sqrt(a+b)*sqrt(sin(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-csc(c+d*x))/(a+b))*sqrt(a*(1+csc(c+d*x))/(a-b))*tan(c+d*x)/d+a*EllipticPi(sqrt(a+b*sin(c+d*x))/(sqrt(a+b)*sqrt(sin(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-csc(c+d*x))/(a+b))*sqrt(a*(1+csc(c+d*x))/(a-b))*tan(c+d*x)/(b*d)],
[1/(sqrt(sin(c+d*x))*sqrt(a+b*sin(c+d*x))),x,1,-2*EllipticF(sqrt(a+b*sin(c+d*x))/(sqrt(a+b)*sqrt(sin(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-csc(c+d*x))/(a+b))*sqrt(a*(1+csc(c+d*x))/(a-b))*tan(c+d*x)/(a*d)],

# Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x])^n with m symbolic
[(d*sin(e+f*x))^m*(a+b*sin(e+f*x))^3,x,5,-a*b^2*(7+2*m)*cos(e+f*x)*(d*sin(e+f*x))^(1+m)/(d*f*(2+m)*(3+m))-b^2*cos(e+f*x)*(d*sin(e+f*x))^(1+m)*(a+b*sin(e+f*x))/(d*f*(3+m))+a*(3*b^2*(1+m)+a^2*(2+m))*cos(e+f*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+m)/(d*f*(1+m)*(2+m)*sqrt(cos(e+f*x)^2))+b*(b^2*(2+m)+3*a^2*(3+m))*cos(e+f*x)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+m)/(d^2*f*(2+m)*(3+m)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^m*(a+b*sin(e+f*x))^2,x,4,-b^2*cos(e+f*x)*(d*sin(e+f*x))^(1+m)/(d*f*(2+m))+(b^2*(1+m)+a^2*(2+m))*cos(e+f*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+m)/(d*f*(1+m)*(2+m)*sqrt(cos(e+f*x)^2))+2*a*b*cos(e+f*x)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+m)/(d^2*f*(2+m)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^m*(a+b*sin(e+f*x)),x,3,a*cos(e+f*x)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],sin(e+f*x)^2)*(d*sin(e+f*x))^(1+m)/(d*f*(1+m)*sqrt(cos(e+f*x)^2))+b*cos(e+f*x)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],sin(e+f*x)^2)*(d*sin(e+f*x))^(2+m)/(d^2*f*(2+m)*sqrt(cos(e+f*x)^2))],
[(d*sin(e+f*x))^m/(a+b*sin(e+f*x)),x,5,-a*d*AppellF1(1/2,1/2*(1-m),1,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/((a^2-b^2)*f)+b*AppellF1(1/2,-1/2*m,1,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^m/((a^2-b^2)*f*(sin(e+f*x)^2)^(1/2*m))],
[(d*sin(e+f*x))^m/(a+b*sin(e+f*x))^2,x,10,-b^2*AppellF1(1/2,1/2*(-1-m),2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^(1+m)*(sin(e+f*x)^2)^(1/2*(-1-m))/((a^2-b^2)^2*d*f)-a^2*d*AppellF1(1/2,1/2*(1-m),2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/((a^2-b^2)^2*f)+2*a*b*AppellF1(1/2,-1/2*m,2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^m/((a^2-b^2)^2*f*(sin(e+f*x)^2)^(1/2*m))],
[(d*sin(e+f*x))^m/(a+b*sin(e+f*x))^3,x,13,-3*a*b^2*AppellF1(1/2,1/2*(-1-m),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^(1+m)*(sin(e+f*x)^2)^(1/2*(-1-m))/((a^2-b^2)^3*d*f)-a^3*d*AppellF1(1/2,1/2*(1-m),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^(-1+m)*(sin(e+f*x)^2)^(1/2*(1-m))/((a^2-b^2)^3*f)+b^3*AppellF1(1/2,1/2*(-2-m),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^m/((a^2-b^2)^3*f*(sin(e+f*x)^2)^(1/2*m))+3*a^2*b*AppellF1(1/2,-1/2*m,3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*sin(e+f*x))^m/((a^2-b^2)^3*f*(sin(e+f*x)^2)^(1/2*m))],
[sin(c+d*x)^(-1-a^2/(a^2+b^2))*(a+b*sin(c+d*x))^2,x,3,-(a^2+b^2)*cos(c+d*x)/(d*sin(c+d*x)^(a^2/(a^2+b^2)))+2*a*(a^2+b^2)*cos(c+d*x)*hypergeom([1/2,1/2*b^2/(a^2+b^2)],[1/2*(3-a^2/(a^2+b^2))],sin(c+d*x)^2)*sin(c+d*x)^(b^2/(a^2+b^2))/(b*d*sqrt(cos(c+d*x)^2))],
[(1+2*sin(c+d*x))^2/sin(c+d*x)^(6/5),x,3,-5*cos(c+d*x)/(d*sin(c+d*x)^(1/5))+5*cos(c+d*x)*hypergeom([2/5,1/2],[7/5],sin(c+d*x)^2)*sin(c+d*x)^(4/5)/(d*sqrt(cos(c+d*x)^2))],

# Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x])^n with n symbolic
[sin(c+d*x)^m*(a+b*sin(c+d*x))^n,x,0,Unintegrable(sin(c+d*x)^m*(a+b*sin(c+d*x))^n,x)],
[sin(c+d*x)^3*(a+b*sin(c+d*x))^n,x,9,2*a*cos(c+d*x)*(a+b*sin(c+d*x))^(1+n)/(b^2*d*(2+n)*(3+n))-cos(c+d*x)*sin(c+d*x)*(a+b*sin(c+d*x))^(1+n)/(b*d*(3+n))-(a+b)*(2*a^2+b^2*(2+n)^2)*AppellF1(1/2,1/2,-1-n,3/2,1/2*(1-sin(c+d*x)),b*(1-sin(c+d*x))/(a+b))*cos(c+d*x)*(a+b*sin(c+d*x))^n*sqrt(2)/(b^3*d*(2+n)*(3+n)*((a+b*sin(c+d*x))/(a+b))^n*sqrt(1+sin(c+d*x)))+a*(2*a^2+b^2*(4+5*n+n^2))*AppellF1(1/2,1/2,-n,3/2,1/2*(1-sin(c+d*x)),b*(1-sin(c+d*x))/(a+b))*cos(c+d*x)*(a+b*sin(c+d*x))^n*sqrt(2)/(b^3*d*(2+n)*(3+n)*((a+b*sin(c+d*x))/(a+b))^n*sqrt(1+sin(c+d*x)))],
[sin(c+d*x)^2*(a+b*sin(c+d*x))^n,x,8,-cos(c+d*x)*(a+b*sin(c+d*x))^(1+n)/(b*d*(2+n))+a*(a+b)*AppellF1(1/2,1/2,-1-n,3/2,1/2*(1-sin(c+d*x)),b*(1-sin(c+d*x))/(a+b))*cos(c+d*x)*(a+b*sin(c+d*x))^n*sqrt(2)/(b^2*d*(2+n)*((a+b*sin(c+d*x))/(a+b))^n*sqrt(1+sin(c+d*x)))-(a^2+b^2*(1+n))*AppellF1(1/2,1/2,-n,3/2,1/2*(1-sin(c+d*x)),b*(1-sin(c+d*x))/(a+b))*cos(c+d*x)*(a+b*sin(c+d*x))^n*sqrt(2)/(b^2*d*(2+n)*((a+b*sin(c+d*x))/(a+b))^n*sqrt(1+sin(c+d*x)))],
[sin(c+d*x)*(a+b*sin(c+d*x))^n,x,7,-(a+b)*AppellF1(1/2,1/2,-1-n,3/2,1/2*(1-sin(c+d*x)),b*(1-sin(c+d*x))/(a+b))*cos(c+d*x)*(a+b*sin(c+d*x))^n*sqrt(2)/(b*d*((a+b*sin(c+d*x))/(a+b))^n*sqrt(1+sin(c+d*x)))+a*AppellF1(1/2,1/2,-n,3/2,1/2*(1-sin(c+d*x)),b*(1-sin(c+d*x))/(a+b))*cos(c+d*x)*(a+b*sin(c+d*x))^n*sqrt(2)/(b*d*((a+b*sin(c+d*x))/(a+b))^n*sqrt(1+sin(c+d*x)))],
[(a+b*sin(c+d*x))^n,x,3,-AppellF1(1/2,1/2,-n,3/2,1/2*(1-sin(c+d*x)),b*(1-sin(c+d*x))/(a+b))*cos(c+d*x)*(a+b*sin(c+d*x))^n*sqrt(2)/(d*((a+b*sin(c+d*x))/(a+b))^n*sqrt(1+sin(c+d*x)))],
[csc(c+d*x)*(a+b*sin(c+d*x))^n,x,0,Unintegrable(csc(c+d*x)*(a+b*sin(c+d*x))^n,x)],

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

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

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

# m>0
[(a+a*sin(e+f*x))*(c-c*sin(e+f*x))^4,x,6,7/8*a*c^4*x+7/12*a*c^4*cos(e+f*x)^3/f+7/8*a*c^4*cos(e+f*x)*sin(e+f*x)/f+1/5*a*cos(e+f*x)^3*(c^2-c^2*sin(e+f*x))^2/f+7/20*a*cos(e+f*x)^3*(c^4-c^4*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(c-c*sin(e+f*x))^3,x,5,5/8*a*c^3*x+5/12*a*c^3*cos(e+f*x)^3/f+5/8*a*c^3*cos(e+f*x)*sin(e+f*x)/f+1/4*a*cos(e+f*x)^3*(c^3-c^3*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(c-c*sin(e+f*x))^2,x,4,1/2*a*c^2*x+1/3*a*c^2*cos(e+f*x)^3/f+1/2*a*c^2*cos(e+f*x)*sin(e+f*x)/f],
[(a+a*sin(e+f*x))*(c-c*sin(e+f*x)),x,1,1/2*a*c*x+1/2*a*c*cos(e+f*x)*sin(e+f*x)/f],
[(a+a*sin(e+f*x))/(c-c*sin(e+f*x)),x,2,-a*x/c+2*a*cos(e+f*x)/(f*(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^2,x,2,1/3*a*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^3)],
[(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^3,x,3,1/5*a*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^4)+1/15*a*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^3)],
[(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^4,x,4,1/7*a*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^5)+2/35*a*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^4)+2/105*a*cos(e+f*x)^3/(c*f*(c-c*sin(e+f*x))^3)],
[(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^5,x,5,1/9*a*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^6)+1/21*a*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^5)+2/105*a*cos(e+f*x)^3/(c*f*(c-c*sin(e+f*x))^4)+2/315*a*c*cos(e+f*x)^3/(f*(c^2-c^2*sin(e+f*x))^3)],
[(a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^5,x,7,9/16*a^2*c^5*x+3/10*a^2*c^5*cos(e+f*x)^5/f+9/16*a^2*c^5*cos(e+f*x)*sin(e+f*x)/f+3/8*a^2*c^5*cos(e+f*x)^3*sin(e+f*x)/f+1/7*a^2*c^3*cos(e+f*x)^5*(c-c*sin(e+f*x))^2/f+3/14*a^2*cos(e+f*x)^5*(c^5-c^5*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^4,x,6,7/16*a^2*c^4*x+7/30*a^2*c^4*cos(e+f*x)^5/f+7/16*a^2*c^4*cos(e+f*x)*sin(e+f*x)/f+7/24*a^2*c^4*cos(e+f*x)^3*sin(e+f*x)/f+1/6*a^2*cos(e+f*x)^5*(c^4-c^4*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^3,x,5,3/8*a^2*c^3*x+1/5*a^2*c^3*cos(e+f*x)^5/f+3/8*a^2*c^3*cos(e+f*x)*sin(e+f*x)/f+1/4*a^2*c^3*cos(e+f*x)^3*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^2,x,4,3/8*a^2*c^2*x+3/8*a^2*c^2*cos(e+f*x)*sin(e+f*x)/f+1/4*a^2*c^2*cos(e+f*x)^3*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^2*(c-c*sin(e+f*x)),x,4,1/2*a^2*c*x-1/3*a^2*c*cos(e+f*x)^3/f+1/2*a^2*c*cos(e+f*x)*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x)),x,4,-3*a^2*x/c+3*a^2*cos(e+f*x)/(c*f)+2*a^2*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^2)],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x))^2,x,4,a^2*x/c^2+2/3*a^2*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^3)-2*a^2*cos(e+f*x)/(f*(c^2-c^2*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x))^3,x,2,1/5*a^2*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^5)],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x))^4,x,3,1/7*a^2*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^6)+1/35*a^2*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^5)],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x))^5,x,4,1/9*a^2*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^7)+2/63*a^2*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^6)+2/315*a^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^5)],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x))^6,x,5,1/11*a^2*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^8)+1/33*a^2*c*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^7)+2/231*a^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^6)+2/1155*a^2*cos(e+f*x)^5/(c*f*(c-c*sin(e+f*x))^5)],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^6,x,8,55/128*a^3*c^6*x+11/56*a^3*c^6*cos(e+f*x)^7/f+55/128*a^3*c^6*cos(e+f*x)*sin(e+f*x)/f+55/192*a^3*c^6*cos(e+f*x)^3*sin(e+f*x)/f+11/48*a^3*c^6*cos(e+f*x)^5*sin(e+f*x)/f+1/9*a^3*cos(e+f*x)^7*(c^3-c^3*sin(e+f*x))^2/f+11/72*a^3*cos(e+f*x)^7*(c^6-c^6*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^5,x,7,45/128*a^3*c^5*x+9/56*a^3*c^5*cos(e+f*x)^7/f+45/128*a^3*c^5*cos(e+f*x)*sin(e+f*x)/f+15/64*a^3*c^5*cos(e+f*x)^3*sin(e+f*x)/f+3/16*a^3*c^5*cos(e+f*x)^5*sin(e+f*x)/f+1/8*a^3*cos(e+f*x)^7*(c^5-c^5*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^4,x,6,5/16*a^3*c^4*x+1/7*a^3*c^4*cos(e+f*x)^7/f+5/16*a^3*c^4*cos(e+f*x)*sin(e+f*x)/f+5/24*a^3*c^4*cos(e+f*x)^3*sin(e+f*x)/f+1/6*a^3*c^4*cos(e+f*x)^5*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^3,x,5,5/16*a^3*c^3*x+5/16*a^3*c^3*cos(e+f*x)*sin(e+f*x)/f+5/24*a^3*c^3*cos(e+f*x)^3*sin(e+f*x)/f+1/6*a^3*c^3*cos(e+f*x)^5*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^2,x,5,3/8*a^3*c^2*x-1/5*a^3*c^2*cos(e+f*x)^5/f+3/8*a^3*c^2*cos(e+f*x)*sin(e+f*x)/f+1/4*a^3*c^2*cos(e+f*x)^3*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x)),x,5,5/8*a^3*c*x-5/12*a^3*c*cos(e+f*x)^3/f+5/8*a^3*c*cos(e+f*x)*sin(e+f*x)/f-1/4*c*cos(e+f*x)^3*(a^3+a^3*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x)),x,5,-15/2*a^3*x/c+15/2*a^3*cos(e+f*x)/(c*f)+2*a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^3)+5/2*a^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^2,x,5,5*a^3*x/c^2-5*a^3*cos(e+f*x)/(c^2*f)+2/3*a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^4)-10/3*a^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^2)],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^3,x,5,-a^3*x/c^3+2/5*a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^5)-2/3*a^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^3)+2*a^3*cos(e+f*x)/(f*(c^3-c^3*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^4,x,2,1/7*a^3*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^7)],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^5,x,3,1/9*a^3*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^8)+1/63*a^3*c^2*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^7)],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^6,x,4,1/11*a^3*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^9)+2/99*a^3*c^2*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^8)+2/693*a^3*c*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^7)],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^7,x,5,1/13*a^3*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^10)+3/143*a^3*c^2*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^9)+2/429*a^3*c*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^8)+2/3003*a^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^7)],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^8,x,6,1/15*a^3*c^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^11)+4/195*a^3*c^2*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^10)+4/715*a^3*c*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^9)+8/6435*a^3*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^8)+8/45045*a^3*cos(e+f*x)^7/(c*f*(c-c*sin(e+f*x))^7)],

# m<0
[(c-c*sin(e+f*x))^4/(a+a*sin(e+f*x)),x,6,-35/2*c^4*x/a-35/3*c^4*cos(e+f*x)^3/(a*f)-35/2*c^4*cos(e+f*x)*sin(e+f*x)/(a*f)-2*a^3*c^4*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^4)-14*a*c^4*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^2)],
[(c-c*sin(e+f*x))^3/(a+a*sin(e+f*x)),x,5,-15/2*c^3*x/a-15/2*c^3*cos(e+f*x)/(a*f)-2*a^2*c^3*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^3)-5/2*c^3*cos(e+f*x)^3/(f*(a+a*sin(e+f*x)))],
[(c-c*sin(e+f*x))^2/(a+a*sin(e+f*x)),x,4,-3*c^2*x/a-3*c^2*cos(e+f*x)/(a*f)-2*a*c^2*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^2)],
[(c-c*sin(e+f*x))/(a+a*sin(e+f*x)),x,2,-c*x/a-2*c*cos(e+f*x)/(f*(a+a*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))),x,3,tan(e+f*x)/(a*c*f)],
[1/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^2),x,4,1/3*sec(e+f*x)/(a*f*(c^2-c^2*sin(e+f*x)))+2/3*tan(e+f*x)/(a*c^2*f)],
[1/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^3),x,5,1/5*sec(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^2)+1/5*sec(e+f*x)/(a*f*(c^3-c^3*sin(e+f*x)))+2/5*tan(e+f*x)/(a*c^3*f)],
[1/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^4),x,6,1/7*sec(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^3)+4/35*sec(e+f*x)/(a*f*(c^2-c^2*sin(e+f*x))^2)+4/35*sec(e+f*x)/(a*f*(c^4-c^4*sin(e+f*x)))+8/35*tan(e+f*x)/(a*c^4*f)],
[(c-c*sin(e+f*x))^5/(a+a*sin(e+f*x))^2,x,7,105/2*c^5*x/a^2+35*c^5*cos(e+f*x)^3/(a^2*f)+105/2*c^5*cos(e+f*x)*sin(e+f*x)/(a^2*f)-2/3*a^4*c^5*cos(e+f*x)^9/(f*(a+a*sin(e+f*x))^6)+6*a^2*c^5*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^4)+42*c^5*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^2)],
[(c-c*sin(e+f*x))^4/(a+a*sin(e+f*x))^2,x,6,35/2*c^4*x/a^2+35/2*c^4*cos(e+f*x)/(a^2*f)-2/3*a^3*c^4*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^5)+14/3*a^4*c^4*cos(e+f*x)^5/(f*(a^2+a^2*sin(e+f*x))^3)+35/6*c^4*cos(e+f*x)^3/(f*(a^2+a^2*sin(e+f*x)))],
[(c-c*sin(e+f*x))^3/(a+a*sin(e+f*x))^2,x,5,5*c^3*x/a^2+5*c^3*cos(e+f*x)/(a^2*f)-2/3*a^2*c^3*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^4)+10/3*c^3*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^2)],
[(c-c*sin(e+f*x))^2/(a+a*sin(e+f*x))^2,x,4,c^2*x/a^2-2/3*a*c^2*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^3)+2*c^2*cos(e+f*x)/(f*(a^2+a^2*sin(e+f*x)))],
[(c-c*sin(e+f*x))/(a+a*sin(e+f*x))^2,x,2,-1/3*a*c*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^3)],
[1/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))),x,4,-1/3*sec(e+f*x)/(c*f*(a^2+a^2*sin(e+f*x)))+2/3*tan(e+f*x)/(a^2*c*f)],
[1/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^2),x,3,tan(e+f*x)/(a^2*c^2*f)+1/3*tan(e+f*x)^3/(a^2*c^2*f)],
[1/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^3),x,4,1/5*sec(e+f*x)^3/(a^2*f*(c^3-c^3*sin(e+f*x)))+4/5*tan(e+f*x)/(a^2*c^3*f)+4/15*tan(e+f*x)^3/(a^2*c^3*f)],
[1/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^4),x,5,1/7*sec(e+f*x)^3/(a^2*f*(c^2-c^2*sin(e+f*x))^2)+1/7*sec(e+f*x)^3/(a^2*f*(c^4-c^4*sin(e+f*x)))+4/7*tan(e+f*x)/(a^2*c^4*f)+4/21*tan(e+f*x)^3/(a^2*c^4*f)],
[1/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^5),x,6,1/9*sec(e+f*x)^3/(a^2*c^2*f*(c-c*sin(e+f*x))^3)+2/21*sec(e+f*x)^3/(a^2*c^3*f*(c-c*sin(e+f*x))^2)+2/21*sec(e+f*x)^3/(a^2*f*(c^5-c^5*sin(e+f*x)))+8/21*tan(e+f*x)/(a^2*c^5*f)+8/63*tan(e+f*x)^3/(a^2*c^5*f)],
[(c-c*sin(e+f*x))^5/(a+a*sin(e+f*x))^3,x,7,-63/2*c^5*x/a^3-63/2*c^5*cos(e+f*x)/(a^3*f)-2/5*a^4*c^5*cos(e+f*x)^9/(f*(a+a*sin(e+f*x))^7)+6/5*a^2*c^5*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^5)-42/5*c^5*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^3)-21/2*c^5*cos(e+f*x)^3/(f*(a^3+a^3*sin(e+f*x)))],
[(c-c*sin(e+f*x))^4/(a+a*sin(e+f*x))^3,x,6,-7*c^4*x/a^3-7*c^4*cos(e+f*x)/(a^3*f)-2/5*a^3*c^4*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^6)+14/15*a*c^4*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^4)-14/3*c^4*cos(e+f*x)^3/(a*f*(a+a*sin(e+f*x))^2)],
[(c-c*sin(e+f*x))^3/(a+a*sin(e+f*x))^3,x,5,-c^3*x/a^3-2/5*a^2*c^3*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^5)+2/3*c^3*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^3)-2*c^3*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))],
[(c-c*sin(e+f*x))^2/(a+a*sin(e+f*x))^3,x,2,-1/5*a^2*c^2*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^5)],
[(c-c*sin(e+f*x))/(a+a*sin(e+f*x))^3,x,3,-1/5*a*c*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^4)-1/15*c*cos(e+f*x)^3/(f*(a+a*sin(e+f*x))^3)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))),x,5,-1/5*sec(e+f*x)/(a*c*f*(a+a*sin(e+f*x))^2)-1/5*sec(e+f*x)/(c*f*(a^3+a^3*sin(e+f*x)))+2/5*tan(e+f*x)/(a^3*c*f)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^2),x,4,-1/5*sec(e+f*x)^3/(c^2*f*(a^3+a^3*sin(e+f*x)))+4/5*tan(e+f*x)/(a^3*c^2*f)+4/15*tan(e+f*x)^3/(a^3*c^2*f)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^3),x,3,tan(e+f*x)/(a^3*c^3*f)+2/3*tan(e+f*x)^3/(a^3*c^3*f)+1/5*tan(e+f*x)^5/(a^3*c^3*f)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^4),x,4,1/7*sec(e+f*x)^5/(a^3*f*(c^4-c^4*sin(e+f*x)))+6/7*tan(e+f*x)/(a^3*c^4*f)+4/7*tan(e+f*x)^3/(a^3*c^4*f)+6/35*tan(e+f*x)^5/(a^3*c^4*f)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^5),x,5,1/9*sec(e+f*x)^5/(a^3*c^3*f*(c-c*sin(e+f*x))^2)+1/9*sec(e+f*x)^5/(a^3*f*(c^5-c^5*sin(e+f*x)))+2/3*tan(e+f*x)/(a^3*c^5*f)+4/9*tan(e+f*x)^3/(a^3*c^5*f)+2/15*tan(e+f*x)^5/(a^3*c^5*f)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^6),x,6,1/11*sec(e+f*x)^5/(a^3*f*(c^2-c^2*sin(e+f*x))^3)+8/99*sec(e+f*x)^5/(a^3*f*(c^3-c^3*sin(e+f*x))^2)+8/99*sec(e+f*x)^5/(a^3*f*(c^6-c^6*sin(e+f*x)))+16/33*tan(e+f*x)/(a^3*c^6*f)+32/99*tan(e+f*x)^3/(a^3*c^6*f)+16/165*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)

# m>0
[(a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(7/2),x,5,256/315*a*c^5*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+2/9*a*c^2*cos(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/f+64/105*a*c^4*cos(e+f*x)^3/(f*sqrt(c-c*sin(e+f*x)))+8/21*a*c^3*cos(e+f*x)^3*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2),x,4,64/105*a*c^4*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+16/35*a*c^3*cos(e+f*x)^3/(f*sqrt(c-c*sin(e+f*x)))+2/7*a*c^2*cos(e+f*x)^3*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2),x,3,8/15*a*c^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+2/5*a*c^2*cos(e+f*x)^3/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2),x,2,2/3*a*c^2*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))],
[(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(1/2),x,4,2*a*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(f*sqrt(c))-2*a*cos(e+f*x)/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,4,a*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(3/2))-a*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(3/2)*f*sqrt(2))],
[(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,5,1/2*a*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(5/2))-1/8*a*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(3/2))-1/8*a*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))/(c-c*sin(e+f*x))^(7/2),x,6,1/3*a*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(7/2))-1/24*a*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(5/2))-1/32*a*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(3/2))-1/32*a*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*(c-c*sin(e+f*x))^(7/2),x,5,256/1155*a^2*c^6*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))+64/231*a^2*c^5*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(3/2))+8/33*a^2*c^4*cos(e+f*x)^5/(f*sqrt(c-c*sin(e+f*x)))+2/11*a^2*c^3*cos(e+f*x)^5*sqrt(c-c*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(5/2),x,4,64/315*a^2*c^5*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))+16/63*a^2*c^4*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(3/2))+2/9*a^2*c^3*cos(e+f*x)^5/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(3/2),x,3,8/35*a^2*c^4*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))+2/7*a^2*c^3*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(3/2))],
[(a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(1/2),x,2,2/5*a^2*c^3*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x))^(1/2),x,5,-2/3*a^2*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+4*a^2*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(f*sqrt(c))-4*a^2*cos(e+f*x)/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x))^(3/2),x,5,a^2*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(5/2))-3*a^2*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(c^(3/2)*f)+3*a^2*cos(e+f*x)/(c*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2/(c-c*sin(e+f*x))^(5/2),x,5,1/2*a^2*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(7/2))-3/4*a^2*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(3/2))+3/4*a^2*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))^2/(c-c*sin(e+f*x))^(7/2),x,6,1/3*a^2*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(9/2))-1/4*a^2*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(5/2))+1/16*a^2*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(3/2))+1/16*a^2*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/(c-c*sin(e+f*x))^(9/2),x,7,1/4*a^2*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(11/2))-1/8*a^2*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(7/2))+1/64*a^2*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(5/2))+3/256*a^2*cos(e+f*x)/(c^3*f*(c-c*sin(e+f*x))^(3/2))+3/256*a^2*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*(c-c*sin(e+f*x))^(7/2),x,5,256/3003*a^3*c^7*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))+64/429*a^3*c^6*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(5/2))+24/143*a^3*c^5*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(3/2))+2/13*a^3*c^4*cos(e+f*x)^7/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(5/2),x,4,64/693*a^3*c^6*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))+16/99*a^3*c^5*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(5/2))+2/11*a^3*c^4*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(3/2))],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(3/2),x,3,8/63*a^3*c^5*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))+2/9*a^3*c^4*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(5/2))],
[(a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(1/2),x,2,2/7*a^3*c^4*cos(e+f*x)^7/(f*(c-c*sin(e+f*x))^(7/2))],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^(1/2),x,6,-2/5*a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(5/2))-4/3*a^3*c*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))+8*a^3*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(f*sqrt(c))-8*a^3*cos(e+f*x)/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^(3/2),x,6,a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(7/2))+5/3*a^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(3/2))-10*a^3*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))*sqrt(2)/(c^(3/2)*f)+10*a^3*cos(e+f*x)/(c*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^(5/2),x,6,1/2*a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(9/2))-5/4*a^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(5/2))+15/2*a^3*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(5/2)*f*sqrt(2))-15/4*a^3*cos(e+f*x)/(c^2*f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c-c*sin(e+f*x))^(7/2),x,6,1/3*a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(11/2))-5/12*a^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(7/2))+5/8*a^3*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(3/2))-5/8*a^3*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))^3/(c-c*sin(e+f*x))^(9/2),x,7,1/4*a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(13/2))-5/24*a^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(9/2))+5/32*a^3*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(5/2))-5/128*a^3*cos(e+f*x)/(c^3*f*(c-c*sin(e+f*x))^(3/2))-5/128*a^3*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/(c-c*sin(e+f*x))^(11/2),x,8,1/5*a^3*c^2*cos(e+f*x)^5/(f*(c-c*sin(e+f*x))^(15/2))-1/8*a^3*cos(e+f*x)^3/(f*(c-c*sin(e+f*x))^(11/2))+1/16*a^3*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(7/2))-1/128*a^3*cos(e+f*x)/(c^3*f*(c-c*sin(e+f*x))^(5/2))-3/512*a^3*cos(e+f*x)/(c^4*f*(c-c*sin(e+f*x))^(3/2))-3/512*a^3*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(c^(11/2)*f*sqrt(2))],

# m<0
[(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x)),x,5,64/5*c^2*sec(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f)+8/5*c*sec(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f)+2/5*sec(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(a*f)-256/5*c^3*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f)],
[(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x)),x,4,16/3*c*sec(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f)+2/3*sec(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f)-64/3*c^2*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f)],
[(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x)),x,3,2*sec(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f)-8*c*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f)],
[(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x)),x,2,-2*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f)],
[1/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(1/2)),x,4,arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a*f*sqrt(2)*sqrt(c))-sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*c*f)],
[1/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(3/2)),x,5,3/4*cos(e+f*x)/(a*f*(c-c*sin(e+f*x))^(3/2))+3/4*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a*c^(3/2)*f*sqrt(2))-sec(e+f*x)/(a*c*f*sqrt(c-c*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))*(c-c*sin(e+f*x))^(5/2)),x,6,15/32*cos(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^(3/2))+1/4*sec(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^(3/2))+15/32*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a*c^(5/2)*f*sqrt(2))-5/8*sec(e+f*x)/(a*c^2*f*sqrt(c-c*sin(e+f*x)))],
[(c-c*sin(e+f*x))^(9/2)/(a+a*sin(e+f*x))^2,x,6,4096/15*c^3*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^2*f)-1024/5*c^2*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^2*f)+128/5*c*sec(e+f*x)^3*(c-c*sin(e+f*x))^(7/2)/(a^2*f)+32/15*sec(e+f*x)^3*(c-c*sin(e+f*x))^(9/2)/(a^2*f)+2/5*sec(e+f*x)^3*(c-c*sin(e+f*x))^(11/2)/(a^2*c*f)],
[(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^2,x,5,256/3*c^2*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^2*f)-64*c*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^2*f)+8*sec(e+f*x)^3*(c-c*sin(e+f*x))^(7/2)/(a^2*f)+2/3*sec(e+f*x)^3*(c-c*sin(e+f*x))^(9/2)/(a^2*c*f)],
[(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^2,x,4,64/3*c*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^2*f)-16*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^2*f)+2*sec(e+f*x)^3*(c-c*sin(e+f*x))^(7/2)/(a^2*c*f)],
[(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^2,x,3,8/3*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^2*f)-2*sec(e+f*x)^3*(c-c*sin(e+f*x))^(5/2)/(a^2*c*f)],
[(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^2,x,2,-2/3*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^2*c*f)],
[1/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(1/2)),x,5,-1/3*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^2*c^2*f)+1/2*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*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*c*f)],
[1/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(3/2)),x,6,5/8*cos(e+f*x)/(a^2*f*(c-c*sin(e+f*x))^(3/2))+5/8*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^2*c^(3/2)*f*sqrt(2))-5/6*sec(e+f*x)/(a^2*c*f*sqrt(c-c*sin(e+f*x)))-1/3*sec(e+f*x)^3*sqrt(c-c*sin(e+f*x))/(a^2*c^2*f)],
[1/((a+a*sin(e+f*x))^2*(c-c*sin(e+f*x))^(5/2)),x,7,35/64*cos(e+f*x)/(a^2*c*f*(c-c*sin(e+f*x))^(3/2))+7/24*sec(e+f*x)/(a^2*c*f*(c-c*sin(e+f*x))^(3/2))+35/64*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^2*c^(5/2)*f*sqrt(2))-35/48*sec(e+f*x)/(a^2*c^2*f*sqrt(c-c*sin(e+f*x)))-1/3*sec(e+f*x)^3/(a^2*c^2*f*sqrt(c-c*sin(e+f*x)))],
[(c-c*sin(e+f*x))^(9/2)/(a+a*sin(e+f*x))^3,x,6,-4096/15*c^2*sec(e+f*x)^5*(c-c*sin(e+f*x))^(5/2)/(a^3*f)+1024/3*c*sec(e+f*x)^5*(c-c*sin(e+f*x))^(7/2)/(a^3*f)-128*sec(e+f*x)^5*(c-c*sin(e+f*x))^(9/2)/(a^3*f)+32/3*sec(e+f*x)^5*(c-c*sin(e+f*x))^(11/2)/(a^3*c*f)+2/3*sec(e+f*x)^5*(c-c*sin(e+f*x))^(13/2)/(a^3*c^2*f)],
[(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^3,x,5,-256/5*c*sec(e+f*x)^5*(c-c*sin(e+f*x))^(5/2)/(a^3*f)+64*sec(e+f*x)^5*(c-c*sin(e+f*x))^(7/2)/(a^3*f)-24*sec(e+f*x)^5*(c-c*sin(e+f*x))^(9/2)/(a^3*c*f)+2*sec(e+f*x)^5*(c-c*sin(e+f*x))^(11/2)/(a^3*c^2*f)],
[(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^3,x,4,-64/15*sec(e+f*x)^5*(c-c*sin(e+f*x))^(5/2)/(a^3*f)+16/3*sec(e+f*x)^5*(c-c*sin(e+f*x))^(7/2)/(a^3*c*f)-2*sec(e+f*x)^5*(c-c*sin(e+f*x))^(9/2)/(a^3*c^2*f)],
[(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^3,x,3,8/15*sec(e+f*x)^5*(c-c*sin(e+f*x))^(5/2)/(a^3*c*f)-2/3*sec(e+f*x)^5*(c-c*sin(e+f*x))^(7/2)/(a^3*c^2*f)],
[(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^3,x,2,-2/5*sec(e+f*x)^5*(c-c*sin(e+f*x))^(5/2)/(a^3*c^2*f)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(1/2)),x,6,-1/6*sec(e+f*x)^3*(c-c*sin(e+f*x))^(3/2)/(a^3*c^2*f)-1/5*sec(e+f*x)^5*(c-c*sin(e+f*x))^(5/2)/(a^3*c^3*f)+1/4*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*sec(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^3*c*f)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(3/2)),x,7,7/16*cos(e+f*x)/(a^3*f*(c-c*sin(e+f*x))^(3/2))-1/5*sec(e+f*x)^5*(c-c*sin(e+f*x))^(3/2)/(a^3*c^3*f)+7/16*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^3*c^(3/2)*f*sqrt(2))-7/12*sec(e+f*x)/(a^3*c*f*sqrt(c-c*sin(e+f*x)))-7/30*sec(e+f*x)^3*sqrt(c-c*sin(e+f*x))/(a^3*c^2*f)],
[1/((a+a*sin(e+f*x))^3*(c-c*sin(e+f*x))^(5/2)),x,8,63/128*cos(e+f*x)/(a^3*c*f*(c-c*sin(e+f*x))^(3/2))+21/80*sec(e+f*x)/(a^3*c*f*(c-c*sin(e+f*x))^(3/2))+63/128*arctanh(cos(e+f*x)*sqrt(c)/(sqrt(2)*sqrt(c-c*sin(e+f*x))))/(a^3*c^(5/2)*f*sqrt(2))-21/32*sec(e+f*x)/(a^3*c^2*f*sqrt(c-c*sin(e+f*x)))-3/10*sec(e+f*x)^3/(a^3*c^2*f*sqrt(c-c*sin(e+f*x)))-1/5*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)

# m>0
[(a+a*sin(e+f*x))^(1/2)*(c-c*sin(e+f*x))^(7/2),x,1,-1/4*a*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)*(c-c*sin(e+f*x))^(5/2),x,1,-1/3*a*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)*(c-c*sin(e+f*x))^(3/2),x,1,-1/2*a*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)*(c-c*sin(e+f*x))^(1/2),x,1,-a*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)/(c-c*sin(e+f*x))^(1/2),x,3,-a*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*sin(e+f*x))^(1/2)/(c-c*sin(e+f*x))^(3/2),x,1,a*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)/(c-c*sin(e+f*x))^(5/2),x,1,1/2*a*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)/(c-c*sin(e+f*x))^(7/2),x,1,1/3*a*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2),x,2,-1/10*a^2*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*sqrt(a+a*sin(e+f*x)))-1/5*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))^(3/2)*(c-c*sin(e+f*x))^(5/2),x,2,-1/6*a^2*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*sqrt(a+a*sin(e+f*x)))-1/4*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))^(3/2)*(c-c*sin(e+f*x))^(3/2),x,2,-1/3*a^2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))-1/3*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)*(c-c*sin(e+f*x))^(1/2),x,1,1/2*c*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(1/2),x,4,-2*a^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*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)/(c-c*sin(e+f*x))^(3/2),x,4,a*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(f*(c-c*sin(e+f*x))^(3/2))+a^2*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+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(5/2),x,1,1/4*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(5/2))],
[(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(7/2),x,2,1/6*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(7/2))+1/24*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)/(c-c*sin(e+f*x))^(9/2),x,2,-1/12*a^2*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+1/4*a*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(f*(c-c*sin(e+f*x))^(9/2))],
[(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(11/2),x,2,-1/20*a^2*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x)))+1/5*a*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(f*(c-c*sin(e+f*x))^(11/2))],
[(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2),x,3,-1/6*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/f-1/15*a^3*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*sqrt(a+a*sin(e+f*x)))-2/15*a^2*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)*(c-c*sin(e+f*x))^(5/2),x,3,-1/5*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)/f-2/15*a^3*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*sqrt(a+a*sin(e+f*x)))-1/5*a^2*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)*(c-c*sin(e+f*x))^(3/2),x,2,1/6*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*sqrt(c-c*sin(e+f*x)))+1/4*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)*(c-c*sin(e+f*x))^(1/2),x,1,1/3*c*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(1/2),x,5,-1/2*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*sqrt(c-c*sin(e+f*x)))-4*a^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^2*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)/(c-c*sin(e+f*x))^(3/2),x,5,a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(3/2))+4*a^3*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*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)/(c-c*sin(e+f*x))^(5/2),x,5,1/2*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(5/2))-a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*(c-c*sin(e+f*x))^(3/2))-a^3*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))^(5/2)/(c-c*sin(e+f*x))^(7/2),x,1,1/6*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(7/2))],
[(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(9/2),x,2,1/8*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(9/2))+1/48*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)/(c-c*sin(e+f*x))^(11/2),x,3,1/10*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(11/2))+1/40*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(9/2))+1/240*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)/(c-c*sin(e+f*x))^(13/2),x,3,1/6*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*(c-c*sin(e+f*x))^(13/2))+1/60*a^3*cos(e+f*x)/(c^2*f*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x)))-1/15*a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*(c-c*sin(e+f*x))^(11/2))],
[(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(9/2),x,4,-3/28*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(9/2)/f-1/8*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(9/2)/f-1/35*a^4*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(f*sqrt(a+a*sin(e+f*x)))-1/14*a^3*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)*(c-c*sin(e+f*x))^(7/2),x,4,-1/7*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/f-1/7*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2)/f-2/35*a^4*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*sqrt(a+a*sin(e+f*x)))-4/35*a^3*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)*(c-c*sin(e+f*x))^(5/2),x,3,1/6*c*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(3/2)/f+1/15*c^3*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*sqrt(c-c*sin(e+f*x)))+2/15*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)*(c-c*sin(e+f*x))^(3/2),x,2,1/10*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*sqrt(c-c*sin(e+f*x)))+1/5*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)*(c-c*sin(e+f*x))^(1/2),x,1,1/4*c*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(1/2),x,6,-a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(f*sqrt(c-c*sin(e+f*x)))-1/3*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*sqrt(c-c*sin(e+f*x)))-8*a^4*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*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)/(c-c*sin(e+f*x))^(3/2),x,6,a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(3/2))+3/2*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*sqrt(c-c*sin(e+f*x)))+12*a^4*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)))+6*a^3*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)/(c-c*sin(e+f*x))^(5/2),x,6,1/2*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(5/2))-3/2*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(3/2))-6*a^4*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*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)/(c-c*sin(e+f*x))^(7/2),x,6,1/3*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(7/2))-1/2*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(5/2))+a^3*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*(c-c*sin(e+f*x))^(3/2))+a^4*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))^(7/2)/(c-c*sin(e+f*x))^(9/2),x,1,1/8*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(9/2))],
[(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(11/2),x,2,1/10*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(11/2))+1/80*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)/(c-c*sin(e+f*x))^(13/2),x,3,1/12*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(13/2))+1/60*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(11/2))+1/480*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)/(c-c*sin(e+f*x))^(15/2),x,4,1/14*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(f*(c-c*sin(e+f*x))^(15/2))+1/56*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(13/2))+1/280*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c^2*f*(c-c*sin(e+f*x))^(11/2))+1/2240*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)/(c-c*sin(e+f*x))^(17/2),x,4,1/8*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(f*(c-c*sin(e+f*x))^(17/2))-3/56*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(15/2))-1/280*a^4*cos(e+f*x)/(c^3*f*(c-c*sin(e+f*x))^(11/2)*sqrt(a+a*sin(e+f*x)))+1/56*a^3*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*(c-c*sin(e+f*x))^(13/2))],

# m<0
[(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(1/2),x,5,1/2*c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))+4*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*c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(1/2),x,4,2*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)))+c*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^(1/2),x,3,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)))],
[1/((a+a*sin(e+f*x))^(1/2)*(c-c*sin(e+f*x))^(1/2)),x,2,arctanh(sin(e+f*x))*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^(1/2)*(c-c*sin(e+f*x))^(3/2)),x,3,1/2*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/2*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)))],
[1/((a+a*sin(e+f*x))^(1/2)*(c-c*sin(e+f*x))^(5/2)),x,4,1/4*cos(e+f*x)/(f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+1/4*cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/4*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)))],
[(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^(3/2),x,6,-c*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*(a+a*sin(e+f*x))^(3/2))-3/2*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*sqrt(a+a*sin(e+f*x)))-12*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)))-6*c^3*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(3/2),x,5,-c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^(3/2))-4*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*c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(3/2),x,4,-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)))-c*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(f*(a+a*sin(e+f*x))^(3/2))],
[(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^(3/2),x,1,-c*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(1/2)),x,3,-1/2*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))+1/2*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)))],
[1/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)),x,4,-1/2*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2))+1/2*cos(e+f*x)/(a*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/2*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)))],
[1/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)),x,5,-1/2*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2))+3/8*cos(e+f*x)/(a*f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+3/8*cos(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+3/8*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)))],
[(c-c*sin(e+f*x))^(9/2)/(a+a*sin(e+f*x))^(5/2),x,7,2*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/2*c*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(f*(a+a*sin(e+f*x))^(5/2))+3*c^3*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))+24*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)))+12*c^4*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^(5/2),x,6,3/2*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/2*c*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(f*(a+a*sin(e+f*x))^(5/2))+6*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*c^3*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(5/2),x,5,-1/2*c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^(5/2))+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)))+c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*(a+a*sin(e+f*x))^(3/2))],
[(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(5/2),x,1,-1/4*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^(5/2))],
[(c-c*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^(5/2),x,1,-1/2*c*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(1/2)),x,4,-1/4*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)))-1/4*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))+1/4*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)))],
[1/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)),x,5,-1/4*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2))-3/8*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2))+3/8*cos(e+f*x)/(a^2*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+3/8*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)))],
[1/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)),x,6,-1/4*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2))-1/2*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2))+3/8*cos(e+f*x)/(a^2*f*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+3/8*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*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 with m and/or n symbolic
[(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,4,2^(1/2+n)*c*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))],
[(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^3,x,4,-1/7*2^(1/2+m)*a^4*c^3*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],
[(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^2,x,4,-1/5*2^(1/2+m)*a^3*c^2*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],
[(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x)),x,4,-1/3*2^(1/2+m)*a^2*c*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],
[(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x)),x,4,2^(1/2+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)],
[(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^2,x,4,1/3*2^(1/2+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)],
[(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^3,x,4,1/5*2^(1/2+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)],
[(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(5/2),x,3,2*c*cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(3/2)/(f*(5+2*m))+64*c^3*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)))+16*c^2*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*(c-c*sin(e+f*x))^(3/2),x,2,8*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(3+8*m+4*m^2)*sqrt(c-c*sin(e+f*x)))+2*c*cos(e+f*x)*(a+a*sin(e+f*x))^m*sqrt(c-c*sin(e+f*x))/(f*(3+2*m))],
[(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(1/2),x,1,2*c*cos(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/(c-c*sin(e+f*x))^(1/2),x,3,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/(c-c*sin(e+f*x))^(3/2),x,3,1/2*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*f*(1+2*m)*sqrt(c-c*sin(e+f*x)))],
[(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^(5/2),x,3,1/4*cos(e+f*x)*hypergeom([3,1/2+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)))],

#  The same rules should be used to integrate the following two problems: 
[(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^(1/2),x,3,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)))],
[(c+c*sin(e+f*x))^m/(a-a*sin(e+f*x))^(1/2),x,3,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*(c-c*sin(e+f*x))^(-3-m),x,3,cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-3-m)/(f*(5+2*m))+2*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*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*(c-c*sin(e+f*x))^(-2-m),x,2,cos(e+f*x)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m)/(f*(3+2*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*(c-c*sin(e+f*x))^(-1-m),x,1,cos(e+f*x)*(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/(c-c*sin(e+f*x))^m,x,4,2^(1/2-m)*c*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*(c-c*sin(e+f*x))^(1-m),x,4,2^(3/2-m)*c^2*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*(c-c*sin(e+f*x))^(2-m),x,4,2^(5/2-m)*c^3*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))],

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

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

# m>0
[(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^4,x,4,1/8*a*(8*c^4+16*c^3*d+24*c^2*d^2+12*c*d^3+3*d^4)*x-1/30*a*(12*c^4+95*c^3*d+112*c^2*d^2+80*c*d^3+16*d^4)*cos(e+f*x)/f-1/120*a*d*(24*c^3+130*c^2*d+116*c*d^2+45*d^3)*cos(e+f*x)*sin(e+f*x)/f-1/60*a*(12*c^2+35*c*d+16*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^2/f-1/20*a*(4*c+5*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/f-1/5*a*cos(e+f*x)*(c+d*sin(e+f*x))^4/f],
[(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^3,x,3,1/8*a*(8*c^3+12*c^2*d+12*c*d^2+3*d^3)*x-1/6*a*(3*c^3+16*c^2*d+12*c*d^2+4*d^3)*cos(e+f*x)/f-1/24*a*d*(6*c^2+20*c*d+9*d^2)*cos(e+f*x)*sin(e+f*x)/f-1/12*a*(3*c+4*d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/f-1/4*a*cos(e+f*x)*(c+d*sin(e+f*x))^3/f],
[(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^2,x,2,1/2*a*(2*c^2+2*c*d+d^2)*x-2/3*a*(c^2+3*c*d+d^2)*cos(e+f*x)/f-1/6*a*d*(2*c+3*d)*cos(e+f*x)*sin(e+f*x)/f-1/3*a*cos(e+f*x)*(c+d*sin(e+f*x))^2/f],
[(a+a*sin(e+f*x))*(c+d*sin(e+f*x)),x,1,1/2*a*(2*c+d)*x-a*(c+d)*cos(e+f*x)/f-1/2*a*d*cos(e+f*x)*sin(e+f*x)/f],
[a+a*sin(e+f*x),x,2,a*x-a*cos(e+f*x)/f],
[(a+a*sin(e+f*x))/(c+d*sin(e+f*x)),x,4,a*x/d-2*a*(c-d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,5,-a*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x)))+2*a*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c+d)*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,6,a*(2*c-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*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x))^2)-1/2*a*(c-2*d)*cos(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^4,x,7,a*(2*c^2-2*c*d+d^2)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c+d)*(c^2-d^2)^(5/2)*f)-1/3*a*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x))^3)-1/6*a*(2*c-3*d)*cos(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sin(e+f*x))^2)-1/6*a*(c-4*d)*(2*c-d)*cos(e+f*x)/((c-d)^2*(c+d)^3*f*(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^4,x,5,1/16*a^2*(24*c^4+64*c^3*d+84*c^2*d^2+48*c*d^3+11*d^4)*x+1/60*a^2*(4*c^5-48*c^4*d-311*c^3*d^2-448*c^2*d^3-288*c*d^4-64*d^5)*cos(e+f*x)/(d*f)+1/240*a^2*(8*c^4-96*c^3*d-438*c^2*d^2-464*c*d^3-165*d^4)*cos(e+f*x)*sin(e+f*x)/f+1/120*a^2*(4*c^3-48*c^2*d-123*c*d^2-64*d^3)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d*f)+1/120*a^2*(4*c^2-48*c*d-55*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d*f)+1/30*a^2*(c-12*d)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d*f)-1/6*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^5/(d*f)],
[(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^3,x,4,3/8*a^2*(2*c+d)*(2*c^2+3*c*d+2*d^2)*x+1/10*a^2*(c^4-10*c^3*d-44*c^2*d^2-40*c*d^3-12*d^4)*cos(e+f*x)/(d*f)+1/40*a^2*(2*c^3-20*c^2*d-57*c*d^2-30*d^3)*cos(e+f*x)*sin(e+f*x)/f+1/20*a^2*(c^2-10*c*d-12*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d*f)+1/20*a^2*(c-10*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d*f)-1/5*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d*f)],
[(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^2,x,3,1/8*a^2*(12*c^2+16*c*d+7*d^2)*x-1/6*a^2*(12*c^2+16*c*d+7*d^2)*cos(e+f*x)/f-1/24*a^2*(12*c^2+16*c*d+7*d^2)*cos(e+f*x)*sin(e+f*x)/f-1/12*(8*c-d)*d*cos(e+f*x)*(a+a*sin(e+f*x))^2/f-1/4*d^2*cos(e+f*x)*(a+a*sin(e+f*x))^3/(a*f)],
[(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x)),x,2,1/2*a^2*(3*c+2*d)*x-2/3*a^2*(3*c+2*d)*cos(e+f*x)/f-1/6*a^2*(3*c+2*d)*cos(e+f*x)*sin(e+f*x)/f-1/3*d*cos(e+f*x)*(a+a*sin(e+f*x))^2/f],
[(a+a*sin(e+f*x))^2,x,1,3/2*a^2*x-2*a^2*cos(e+f*x)/f-1/2*a^2*cos(e+f*x)*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^2/(c+d*sin(e+f*x)),x,5,-a^2*(c-2*d)*x/d^2-a^2*cos(e+f*x)/(d*f)+2*a^2*(c-d)^2*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))^2/(c+d*sin(e+f*x))^2,x,5,a^2*x/d^2-2*a^2*(c-d)^2*(c+2*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^2*(c^2-d^2)^(3/2)*f)+a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))),a^2*x/d^2+a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x)))-2*a^2*(c-d)*(c+2*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^2*(c+d)*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^2/(c+d*sin(e+f*x))^3,x,6,1/2*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^2)-1/2*a^2*(c+4*d)*cos(e+f*x)/(d*(c+d)^2*f*(c+d*sin(e+f*x)))+3*a^2*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c+d)^2*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^2/(c+d*sin(e+f*x))^4,x,7,1/3*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^3)-1/6*a^2*(c+6*d)*cos(e+f*x)/(d*(c+d)^2*f*(c+d*sin(e+f*x))^2)-1/6*a^2*(c^2+6*c*d-10*d^2)*cos(e+f*x)/((c-d)*d*(c+d)^3*f*(c+d*sin(e+f*x)))+a^2*(3*c-2*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c-d)*(c+d)^3*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^2/(c+d*sin(e+f*x))^5,x,8,1/4*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^4)-1/12*a^2*(c+8*d)*cos(e+f*x)/(d*(c+d)^2*f*(c+d*sin(e+f*x))^3)-1/24*a^2*(2*c^2+16*c*d-21*d^2)*cos(e+f*x)/((c-d)*d*(c+d)^3*f*(c+d*sin(e+f*x))^2)-1/24*a^2*(2*c^3+16*c^2*d-59*c*d^2+32*d^3)*cos(e+f*x)/((c-d)^2*d*(c+d)^4*f*(c+d*sin(e+f*x)))+1/4*a^2*(12*c^2-16*c*d+7*d^2)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c-d)^2*(c+d)^4*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^3,x,6,1/16*a^3*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*x-4*a^3*(c+d)^3*cos(e+f*x)/f+1/3*a^3*(c+d)^2*(c+7*d)*cos(e+f*x)^3/f-3/5*a^3*d^2*(c+d)*cos(e+f*x)^5/f-1/16*a^3*(24*c^3+90*c^2*d+78*c*d^2+23*d^3)*cos(e+f*x)*sin(e+f*x)/f-1/24*a^3*d*(18*c^2+54*c*d+23*d^2)*cos(e+f*x)*sin(e+f*x)^3/f-1/6*a^3*d^3*cos(e+f*x)*sin(e+f*x)^5/f,1/16*a^3*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*x-1/60*a^3*(2*c^5-18*c^4*d+107*c^3*d^2+472*c^2*d^3+456*c*d^4+136*d^5)*cos(e+f*x)/(d^2*f)-1/240*a^3*(4*c^4-36*c^3*d+216*c^2*d^2+626*c*d^3+345*d^4)*cos(e+f*x)*sin(e+f*x)/(d*f)-1/120*a^3*(2*c^3-18*c^2*d+111*c*d^2+136*d^3)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d^2*f)-1/120*a^3*(2*c^2-18*c*d+115*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^2*f)+1/30*a^3*(2*c-13*d)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d^2*f)-1/6*cos(e+f*x)*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x))^4/(d*f)],
[(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^2,x,9,1/8*a^3*(20*c^2+30*c*d+13*d^2)*x-4*a^3*(c+d)^2*cos(e+f*x)/f+1/3*a^3*(c^2+6*c*d+5*d^2)*cos(e+f*x)^3/f-1/5*a^3*d^2*cos(e+f*x)^5/f-1/8*a^3*(12*c^2+30*c*d+13*d^2)*cos(e+f*x)*sin(e+f*x)/f-1/4*a^3*d*(2*c+3*d)*cos(e+f*x)*sin(e+f*x)^3/f,1/8*a^3*(20*c^2+30*c*d+13*d^2)*x-1/5*a^3*(20*c^2+30*c*d+13*d^2)*cos(e+f*x)/f+1/60*a^3*(20*c^2+30*c*d+13*d^2)*cos(e+f*x)^3/f-3/40*a^3*(20*c^2+30*c*d+13*d^2)*cos(e+f*x)*sin(e+f*x)/f-1/20*(10*c-d)*d*cos(e+f*x)*(a+a*sin(e+f*x))^3/f-1/5*d^2*cos(e+f*x)*(a+a*sin(e+f*x))^4/(a*f)],
[(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x)),x,8,5/8*a^3*(4*c+3*d)*x-4*a^3*(c+d)*cos(e+f*x)/f+1/3*a^3*(c+3*d)*cos(e+f*x)^3/f-3/8*a^3*(4*c+5*d)*cos(e+f*x)*sin(e+f*x)/f-1/4*a^3*d*cos(e+f*x)*sin(e+f*x)^3/f,5/8*a^3*(4*c+3*d)*x-a^3*(4*c+3*d)*cos(e+f*x)/f+1/12*a^3*(4*c+3*d)*cos(e+f*x)^3/f-3/8*a^3*(4*c+3*d)*cos(e+f*x)*sin(e+f*x)/f-1/4*d*cos(e+f*x)*(a+a*sin(e+f*x))^3/f],
[(a+a*sin(e+f*x))^3,x,7,5/2*a^3*x-4*a^3*cos(e+f*x)/f+1/3*a^3*cos(e+f*x)^3/f-3/2*a^3*cos(e+f*x)*sin(e+f*x)/f],
[(a+a*sin(e+f*x))^3/(c+d*sin(e+f*x)),x,7,1/2*a^3*(2*c^2-6*c*d+7*d^2)*x/d^3+1/2*a^3*(2*c-5*d)*cos(e+f*x)/(d^2*f)-1/2*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*f)-2*a^3*(c-d)^3*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))^3/(c+d*sin(e+f*x))^2,x,7,-a^3*(2*c-3*d)*x/d^3-2*a^3*c*cos(e+f*x)/(d^2*(c+d)*f)+(c-d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x)))+2*a^3*(c-d)^2*(2*c+3*d)*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))^3/(c+d*sin(e+f*x))^3,x,7,a^3*x/d^3+1/2*(c-d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^2)+1/2*a^3*(c-d)*(2*c+5*d)*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x)))-a^3*(c-d)*(2*c^2+6*c*d+7*d^2)*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/(c+d*sin(e+f*x))^4,x,8,1/3*(c-d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^3)+1/6*a^3*(c-d)*(2*c+7*d)*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x))^2)-1/6*a^3*(2*c^2+9*c*d+22*d^2)*cos(e+f*x)/(d^2*(c+d)^3*f*(c+d*sin(e+f*x)))+5*a^3*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c+d)^3*f*sqrt(c^2-d^2))],
[(a+a*sin(e+f*x))^3/(c+d*sin(e+f*x))^5,x,9,1/4*(c-d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^4)+1/12*a^3*(c-d)*(2*c+9*d)*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x))^3)-1/24*a^3*(2*c^2+12*c*d+45*d^2)*cos(e+f*x)/(d^2*(c+d)^3*f*(c+d*sin(e+f*x))^2)-1/24*a^3*(2*c^3+12*c^2*d+43*c*d^2-72*d^3)*cos(e+f*x)/((c-d)*d^2*(c+d)^4*f*(c+d*sin(e+f*x)))+5/4*a^3*(4*c-3*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c-d)*(c+d)^4*f*sqrt(c^2-d^2))],

# m<0
[(c+d*sin(e+f*x))^4/(a+a*sin(e+f*x)),x,3,1/2*d*(8*c^3-12*c^2*d+12*c*d^2-3*d^3)*x/a+2/3*d*(3*c^3-16*c^2*d+12*c*d^2-4*d^3)*cos(e+f*x)/(a*f)+1/6*d^2*(6*c^2-20*c*d+9*d^2)*cos(e+f*x)*sin(e+f*x)/(a*f)+1/3*(3*c-4*d)*d*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a*f)-(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a+a*sin(e+f*x)))],
[(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x)),x,2,3/2*d*(2*c^2-2*c*d+d^2)*x/a+2*d*(c^2-3*c*d+d^2)*cos(e+f*x)/(a*f)+1/2*(2*c-3*d)*d^2*cos(e+f*x)*sin(e+f*x)/(a*f)-(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x)))],
[(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x)),x,3,(2*c-d)*d*x/a-d^2*cos(e+f*x)/(a*f)-(c-d)^2*cos(e+f*x)/(a*f*(1+sin(e+f*x)))],
[(c+d*sin(e+f*x))/(a+a*sin(e+f*x)),x,2,d*x/a-(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x)))],
[1/(a+a*sin(e+f*x)),x,1,-cos(e+f*x)/(f*(a+a*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))),x,5,-cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x)))-2*d*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(a*(c-d)*f*sqrt(c^2-d^2))],
[1/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))^2),x,6,-2*d*(2*c+d)*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*(c+2*d)*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sin(e+f*x)))-cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))*(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))^3),x,7,-3*d*(2*c^2+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)^(5/2)*f)-1/2*d*(2*c+3*d)*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sin(e+f*x))^2)-cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^2)-1/2*d*(2*c+d)*(c+4*d)*cos(e+f*x)/(a*(c-d)^3*(c+d)^2*f*(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^5/(a+a*sin(e+f*x))^2,x,4,5/2*(2*c-d)*d^2*(2*c^2-3*c*d+2*d^2)*x/a^2+2/3*d*(c^4+10*c^3*d-44*c^2*d^2+40*c*d^3-12*d^4)*cos(e+f*x)/(a^2*f)+1/6*d^2*(2*c^3+20*c^2*d-57*c*d^2+30*d^3)*cos(e+f*x)*sin(e+f*x)/(a^2*f)+1/3*d*(c^2+10*c*d-12*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a^2*f)-1/3*(c-d)*(c+10*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(a^2*f*(1+sin(e+f*x)))-1/3*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(f*(a+a*sin(e+f*x))^2)],
[(c+d*sin(e+f*x))^4/(a+a*sin(e+f*x))^2,x,3,1/2*d^2*(12*c^2-16*c*d+7*d^2)*x/a^2+2/3*d*(c^3+8*c^2*d-20*c*d^2+8*d^3)*cos(e+f*x)/(a^2*f)+1/6*d^2*(2*c^2+16*c*d-21*d^2)*cos(e+f*x)*sin(e+f*x)/(a^2*f)-1/3*(c-d)*(c+8*d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a^2*f*(1+sin(e+f*x)))-1/3*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a+a*sin(e+f*x))^2)],
[(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^2,x,5,(3*c-2*d)*d^2*x/a^2+1/3*(c-4*d)*d^2*cos(e+f*x)/(a^2*f)-1/3*(c-d)^2*(c+6*d)*cos(e+f*x)/(a^2*f*(1+sin(e+f*x)))-1/3*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x))^2)],
[(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^2,x,3,d^2*x/a^2-1/3*(c-d)*(c+4*d)*cos(e+f*x)/(a^2*f*(1+sin(e+f*x)))-1/3*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^2)],
[(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^2,x,2,-1/3*(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^2)-1/3*(c+2*d)*cos(e+f*x)/(f*(a^2+a^2*sin(e+f*x)))],
[1/(a+a*sin(e+f*x))^2,x,2,-1/3*cos(e+f*x)/(f*(a+a*sin(e+f*x))^2)-1/3*cos(e+f*x)/(f*(a^2+a^2*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))),x,6,-1/3*(c-4*d)*cos(e+f*x)/(a^2*(c-d)^2*f*(1+sin(e+f*x)))-1/3*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^2)+2*d^2*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))],
[1/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^2),x,7,-1/3*d*(c^2-6*c*d-10*d^2)*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sin(e+f*x)))-1/3*(c-6*d)*cos(e+f*x)/(a^2*(c-d)^2*f*(1+sin(e+f*x))*(c+d*sin(e+f*x)))-1/3*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x)))+2*d^2*(3*c+2*d)*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))],
[1/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^3),x,8,-1/6*d*(2*c^2-16*c*d-21*d^2)*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sin(e+f*x))^2)-1/3*(c-8*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*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^2)-1/6*d*(2*c^3-16*c^2*d-59*c*d^2-32*d^3)*cos(e+f*x)/(a^2*(c-d)^4*(c+d)^2*f*(c+d*sin(e+f*x)))+d^2*(12*c^2+16*c*d+7*d^2)*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))],
[(c+d*sin(e+f*x))^6/(a+a*sin(e+f*x))^3,x,5,1/2*d^3*(40*c^3-90*c^2*d+78*c*d^2-23*d^3)*x/a^3+2/15*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)*cos(e+f*x)/(a^3*f)+1/30*d^2*(4*c^4+36*c^3*d+216*c^2*d^2-626*c*d^3+345*d^4)*cos(e+f*x)*sin(e+f*x)/(a^3*f)+1/15*d*(2*c^3+18*c^2*d+111*c*d^2-136*d^3)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a^3*f)-1/15*(c-d)*(2*c^2+18*c*d+115*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a^3+a^3*sin(e+f*x)))-1/15*(c-d)*(2*c+13*d)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(a*f*(a+a*sin(e+f*x))^2)-1/5*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^5/(f*(a+a*sin(e+f*x))^3)],
[(c+d*sin(e+f*x))^5/(a+a*sin(e+f*x))^3,x,4,1/2*d^3*(20*c^2-30*c*d+13*d^2)*x/a^3+2/15*d*(2*c^4+15*c^3*d+72*c^2*d^2-180*c*d^3+76*d^4)*cos(e+f*x)/(a^3*f)+1/30*d^2*(4*c^3+30*c^2*d+146*c*d^2-195*d^3)*cos(e+f*x)*sin(e+f*x)/(a^3*f)-1/15*(c-d)*(2*c^2+15*c*d+76*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a^3+a^3*sin(e+f*x)))-1/15*(c-d)*(2*c+11*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(a*f*(a+a*sin(e+f*x))^2)-1/5*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(f*(a+a*sin(e+f*x))^3)],
[(c+d*sin(e+f*x))^4/(a+a*sin(e+f*x))^3,x,6,(4*c-3*d)*d^3*x/a^3+1/15*d^2*(2*c^2+10*c*d-27*d^2)*cos(e+f*x)/(a^3*f)-1/15*(c-d)^2*(2*c^2+12*c*d+45*d^2)*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))-1/15*(c-d)*(2*c+9*d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(a*f*(a+a*sin(e+f*x))^2)-1/5*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*(a+a*sin(e+f*x))^3)],
[(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^3,x,5,d^3*x/a^3-1/15*(c-d)^2*(2*c+7*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^2)-1/15*(c-d)*(2*c^2+11*c*d+29*d^2)*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))-1/5*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x))^3)],
[(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^3,x,3,-1/15*(c-d)*(2*c+5*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^2)-1/15*(2*c^2+6*c*d+7*d^2)*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))-1/5*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^3)],
[(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^3,x,3,-1/5*(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^3)-1/15*(2*c+3*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^2)-1/15*(2*c+3*d)*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))],
[1/(a+a*sin(e+f*x))^3,x,3,-1/5*cos(e+f*x)/(f*(a+a*sin(e+f*x))^3)-2/15*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^2)-2/15*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))),x,7,-1/5*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^3)-1/15*(2*c-7*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^2)-1/15*(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^3*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))],
[1/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^2),x,8,-1/15*d*(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*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x)))-1/15*(2*c-9*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*(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^3*(4*c+3*d)*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))],
[1/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^3),x,9,-1/30*d*(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*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^2)-1/15*(2*c-11*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*(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*(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^3*(20*c^2+30*c*d+13*d^2)*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))],
[(A+B*sin(x))/(1+sin(x))^4,x,4,-1/7*(A-B)*cos(x)/(1+sin(x))^4-1/35*(3*A+4*B)*cos(x)/(1+sin(x))^3-2/105*(3*A+4*B)*cos(x)/(1+sin(x))^2-2/105*(3*A+4*B)*cos(x)/(1+sin(x))],
[(A+B*sin(x))/(1-sin(x))^4,x,4,1/7*(A+B)*cos(x)/(1-sin(x))^4+1/35*(3*A-4*B)*cos(x)/(1-sin(x))^3+2/105*(3*A-4*B)*cos(x)/(1-sin(x))^2+2/105*(3*A-4*B)*cos(x)/(1-sin(x))],

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

# m>0
[(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^(5/2),x,8,-2/35*a*(5*c+7*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/f-2/7*a*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/f-2/105*a*(15*c^2+56*c*d+25*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/f+2/105*a*(15*c^3+161*c^2*d+145*c*d^2+63*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/105*a*(c^2-d^2)*(15*c^2+56*c*d+25*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2),x,7,-2/5*a*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/f-2/15*a*(3*c+5*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/f+2/15*a*(3*c^2+20*c*d+9*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/15*a*(3*c+5*d)*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^(1/2),x,6,-2/3*a*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/f+2/3*a*(c+3*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/3*a*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^(1/2),x,5,2*a*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2*a*(c-d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^(3/2),x,6,-2*a*cos(e+f*x)/((c+d)*f*sqrt(c+d*sin(e+f*x)))-2*a*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*(c+d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2*a*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^(5/2),x,7,-2/3*a*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x))^(3/2))-2/3*a*(c-3*d)*cos(e+f*x)/((c-d)*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))-2/3*a*(c-3*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((c-d)*d*(c+d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/3*a*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*(c+d)*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))/(c+d*sin(e+f*x))^(7/2),x,8,-2/5*a*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x))^(5/2))-2/15*a*(3*c-5*d)*cos(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sin(e+f*x))^(3/2))-2/15*a*(3*c^2-20*c*d+9*d^2)*cos(e+f*x)/((c-d)^2*(c+d)^3*f*sqrt(c+d*sin(e+f*x)))-2/15*a*(3*c^2-20*c*d+9*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((c-d)^2*d*(c+d)^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/15*a*(3*c-5*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((c-d)*d*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(5/2),x,9,4/315*a^2*(5*c*(c-9*d)-56*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f)+4/63*a^2*(c-9*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d*f)-2/9*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(7/2)/(d*f)+4/315*a^2*(5*c^3-45*c^2*d-141*c*d^2-75*d^3)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f)-4/315*a^2*(5*c^4-45*c^3*d-381*c^2*d^2-435*c*d^3-168*d^4)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/315*a^2*(c^2-d^2)*(5*c^3-45*c^2*d-141*c*d^2-75*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(3/2),x,8,4/35*a^2*(c-7*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f)-2/7*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d*f)+4/35*a^2*(c^2-7*c*d-10*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f)-4/35*a^2*(c+3*d)*(c^2-10*c*d-7*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/35*a^2*(c^2-7*c*d-10*d^2)*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(1/2),x,7,-2/5*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f)+4/15*a^2*(c-5*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f)-4/15*a^2*(c^2-5*c*d-12*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/15*a^2*(c-5*d)*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2/(c+d*sin(e+f*x))^(1/2),x,6,-2/3*a^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f)-4/3*a^2*(c-3*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/3*a^2*(c-2*d)*(c-d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2/(c+d*sin(e+f*x))^(3/2),x,6,2*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*sqrt(c+d*sin(e+f*x)))+4*a^2*c*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*(c+d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-4*a^2*(c-d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2/(c+d*sin(e+f*x))^(5/2),x,7,2/3*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^(3/2))-4/3*a^2*(c+3*d)*cos(e+f*x)/(d*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))-4/3*a^2*(c+3*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*(c+d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/3*a^2*(c+2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*(c+d)*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^2/(c+d*sin(e+f*x))^(7/2),x,8,2/5*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^(5/2))-4/15*a^2*(c+5*d)*cos(e+f*x)/(d*(c+d)^2*f*(c+d*sin(e+f*x))^(3/2))-4/15*a^2*(c^2+5*c*d-12*d^2)*cos(e+f*x)/((c-d)*d*(c+d)^3*f*sqrt(c+d*sin(e+f*x)))-4/15*a^2*(c^2+5*c*d-12*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((c-d)*d^2*(c+d)^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/15*a^2*(c+5*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^(5/2),x,11,-4/693*a^3*(4*c^3-33*c^2*d+182*c*d^2+231*d^3)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f)-4/693*a^3*(4*c^2-33*c*d+189*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d^2*f)+8/99*a^3*(c-6*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(7/2)/(d^2*f)-2/11*cos(e+f*x)*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x))^(7/2)/(d*f)-4/693*a^3*(4*c^4-33*c^3*d+177*c^2*d^2+561*c*d^3+315*d^4)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f)+4/693*a^3*(c+3*d)*(4*c^4-45*c^3*d+309*c^2*d^2+525*c*d^3+231*d^4)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))-4/693*a^3*(c^2-d^2)*(4*c^4-33*c^3*d+177*c^2*d^2+561*c*d^3+315*d^4)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^(3/2),x,10,-4/315*a^3*(4*c^2-27*c*d+119*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f)+8/63*a^3*(c-5*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d^2*f)-2/9*cos(e+f*x)*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x))^(5/2)/(d*f)-4/315*a^3*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f)+4/315*a^3*(4*c^4-27*c^3*d+111*c^2*d^2+579*c*d^3+357*d^4)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))-4/315*a^3*(c^2-d^2)*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^(1/2),x,9,8/35*a^3*(c-4*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f)-2/7*cos(e+f*x)*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2)/(d*f)-4/105*a^3*(4*c^2-21*c*d+65*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f)+4/105*a^3*(4*c^3-21*c^2*d+62*c*d^2+147*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))-4/105*a^3*(c^2-d^2)*(4*c^2-21*c*d+65*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c+d*sin(e+f*x))^(1/2),x,8,8/15*a^3*(c-3*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f)-2/5*cos(e+f*x)*(a^3+a^3*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(d*f)+4/15*a^3*(4*c^2-15*c*d+27*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))-4/15*a^3*(c-d)*(4*c^2-11*c*d+15*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c+d*sin(e+f*x))^(3/2),x,8,2*(c-d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*(c+d)*f*sqrt(c+d*sin(e+f*x)))-4/3*a^3*(2*c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*(c+d)*f)-4/3*a^3*(4*c^2-5*c*d-3*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*(c+d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/3*a^3*(4*c-5*d)*(c-d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c+d*sin(e+f*x))^(5/2),x,8,2/3*(c-d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^(3/2))+8/3*a^3*(c-d)*(c+2*d)*cos(e+f*x)/(d^2*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))+4/3*a^3*(4*c^2+5*c*d-3*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*(c+d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))-4/3*a^3*(c-d)*(4*c+5*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*(c+d)*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c+d*sin(e+f*x))^(7/2),x,9,2/5*(c-d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^(5/2))+8/15*a^3*(c-d)*(c+3*d)*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x))^(3/2))-4/15*a^3*(4*c^2+15*c*d+27*d^2)*cos(e+f*x)/(d^2*(c+d)^3*f*sqrt(c+d*sin(e+f*x)))-4/15*a^3*(4*c^2+15*c*d+27*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*(c+d)^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/15*a^3*(4*c^2+11*c*d+15*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^3/(c+d*sin(e+f*x))^(9/2),x,10,2/7*(c-d)*cos(e+f*x)*(a^3+a^3*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^(7/2))+8/35*a^3*(c-d)*(c+4*d)*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x))^(5/2))-4/105*a^3*(4*c^2+21*c*d+65*d^2)*cos(e+f*x)/(d^2*(c+d)^3*f*(c+d*sin(e+f*x))^(3/2))-4/105*a^3*(4*c^3+21*c^2*d+62*c*d^2-147*d^3)*cos(e+f*x)/((c-d)*d^2*(c+d)^4*f*sqrt(c+d*sin(e+f*x)))-4/105*a^3*(4*c^3+21*c^2*d+62*c*d^2-147*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((c-d)*d^3*(c+d)^4*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/105*a^3*(4*c^2+21*c*d+65*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*(c+d)^3*f*sqrt(c+d*sin(e+f*x)))],

# m<0
[(c+d*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x)),x,7,-(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x)))+1/3*(3*c-5*d)*d*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*f)-1/3*(3*c^2-20*c*d+9*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/3*(3*c-5*d)*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x)),x,6,-(c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x)))-(c-3*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a*f*sqrt((c+d*sin(e+f*x))/(c+d)))+(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x)),x,6,-cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x)))-sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a*f*sqrt((c+d*sin(e+f*x))/(c+d)))+(c+d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))^(1/2)),x,6,-cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((c-d)*f*(a+a*sin(e+f*x)))-sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a*(c-d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2)),x,7,-d*(c+3*d)*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*sqrt(c+d*sin(e+f*x)))-cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))-(c+3*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a*(c-d)^2*(c+d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a*(c-d)*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))*(c+d*sin(e+f*x))^(5/2)),x,8,-1/3*d*(3*c+5*d)*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sin(e+f*x))^(3/2))-cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2))-1/3*d*(3*c^2+20*c*d+9*d^2)*cos(e+f*x)/(a*(c-d)^3*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))-1/3*(3*c^2+20*c*d+9*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a*(c-d)^3*(c+d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/3*(3*c+5*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a*(c-d)^2*(c+d)*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^2,x,7,-1/3*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^2)-1/3*(c-d)*(c+5*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a^2*f*(1+sin(e+f*x)))-1/3*(c^2+5*c*d-12*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/3*(c+5*d)*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^2*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^2,x,7,-1/3*(c+3*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a^2*f*(1+sin(e+f*x)))-1/3*(c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^2)-1/3*(c+3*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/3*(c+d)*(c+2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^2*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^2,x,7,-1/3*c*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a^2*(c-d)*f*(1+sin(e+f*x)))-1/3*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^2)-1/3*c*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^2*(c-d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/3*(c+d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^2*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(1/2)),x,7,-1/3*(c-3*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a^2*(c-d)^2*f*(1+sin(e+f*x)))-1/3*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((c-d)*f*(a+a*sin(e+f*x))^2)-1/3*(c-3*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^2*(c-d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/3*(c-2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^2*(c-d)*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(3/2)),x,8,-1/3*d*(c^2-5*c*d-12*d^2)*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*sqrt(c+d*sin(e+f*x)))-1/3*(c-5*d)*cos(e+f*x)/(a^2*(c-d)^2*f*(1+sin(e+f*x))*sqrt(c+d*sin(e+f*x)))-1/3*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^2*sqrt(c+d*sin(e+f*x)))-1/3*(c^2-5*c*d-12*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^2*(c-d)^3*(c+d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/3*(c-5*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^2*(c-d)^2*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(5/2)),x,9,-1/3*d*(c^2-7*c*d-10*d^2)*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sin(e+f*x))^(3/2))-1/3*(c-7*d)*cos(e+f*x)/(a^2*(c-d)^2*f*(1+sin(e+f*x))*(c+d*sin(e+f*x))^(3/2))-1/3*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(3/2))-1/3*d*(c+3*d)*(c^2-10*c*d-7*d^2)*cos(e+f*x)/(a^2*(c-d)^4*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))-1/3*(c+3*d)*(c^2-10*c*d-7*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^2*(c-d)^4*(c+d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/3*(c^2-7*c*d-10*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^2*(c-d)^3*(c+d)*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^3,x,8,-1/5*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^3)-2/15*(c-d)*(c+3*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*f*(a+a*sin(e+f*x))^2)-1/30*(4*c^2+15*c*d+27*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a^3+a^3*sin(e+f*x)))-1/30*(4*c^2+15*c*d+27*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/30*(c+d)*(4*c^2+11*c*d+15*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^3*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^3,x,8,-1/5*(c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^3)-2/15*(c+2*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*f*(a+a*sin(e+f*x))^2)-1/30*(4*c^2+5*c*d-3*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((c-d)*f*(a^3+a^3*sin(e+f*x)))-1/30*(4*c^2+5*c*d-3*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^3*(c-d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/30*(c+d)*(4*c+5*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^3*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^3,x,8,-1/5*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^3)-1/15*(2*c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*(c-d)*f*(a+a*sin(e+f*x))^2)-1/30*(4*c^2-5*c*d-3*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((c-d)^2*f*(a^3+a^3*sin(e+f*x)))-1/30*(4*c^2-5*c*d-3*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^3*(c-d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/30*(4*c-5*d)*(c+d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^3*(c-d)*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^(1/2)),x,8,-1/5*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((c-d)*f*(a+a*sin(e+f*x))^3)-2/15*(c-3*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*(c-d)^2*f*(a+a*sin(e+f*x))^2)-1/30*(4*c^2-15*c*d+27*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((c-d)^3*f*(a^3+a^3*sin(e+f*x)))-1/30*(4*c^2-15*c*d+27*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^3*(c-d)^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/30*(4*c^2-11*c*d+15*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^3*(c-d)^2*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^(3/2)),x,9,-1/30*d*(4*c^3-21*c^2*d+62*c*d^2+147*d^3)*cos(e+f*x)/(a^3*(c-d)^4*(c+d)*f*sqrt(c+d*sin(e+f*x)))-1/5*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^3*sqrt(c+d*sin(e+f*x)))-2/15*(c-4*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^2*sqrt(c+d*sin(e+f*x)))-1/30*(4*c^2-21*c*d+65*d^2)*cos(e+f*x)/((c-d)^3*f*(a^3+a^3*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))-1/30*(4*c^3-21*c^2*d+62*c*d^2+147*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^3*(c-d)^4*(c+d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/30*(4*c^2-21*c*d+65*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^3*(c-d)^3*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^(5/2)),x,10,-1/30*d*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*cos(e+f*x)/(a^3*(c-d)^4*(c+d)*f*(c+d*sin(e+f*x))^(3/2))-1/5*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^(3/2))-2/15*(c-5*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^(3/2))-1/30*(4*c^2-27*c*d+119*d^2)*cos(e+f*x)/((c-d)^3*f*(a^3+a^3*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2))-1/30*d*(4*c^4-27*c^3*d+111*c^2*d^2+579*c*d^3+357*d^4)*cos(e+f*x)/(a^3*(c-d)^5*(c+d)^2*f*sqrt(c+d*sin(e+f*x)))-1/30*(4*c^4-27*c^3*d+111*c^2*d^2+579*c*d^3+357*d^4)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(a^3*(c-d)^5*(c+d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/30*(4*c^3-27*c^2*d+114*c*d^2+165*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(a^3*(c-d)^4*(c+d)*f*sqrt(c+d*sin(e+f*x)))],

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

# m>0
[(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^3,x,4,-12/35*d^2*(c+d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(a*f)-4/35*a*(c+d)*(15*c^2+10*c*d+7*d^2)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/7*a*cos(e+f*x)*(c+d*sin(e+f*x))^3/(f*sqrt(a+a*sin(e+f*x)))-8/35*(5*c-d)*d*(c+d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^2,x,3,-2/5*d^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(a*f)-2/15*a*(15*c^2+10*c*d+7*d^2)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-4/15*(5*c-d)*d*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x)),x,2,-2/3*a*(3*c+d)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/3*d*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(1/2),x,1,-2*a*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x)),x,2,-2*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(a)/(f*sqrt(d)*sqrt(c+d))],
[(a+a*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^2,x,3,-arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(a)/((c+d)^(3/2)*f*sqrt(d))-a*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^3,x,4,-3/4*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(a)/((c+d)^(5/2)*f*sqrt(d))-1/2*a*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x))^2*sqrt(a+a*sin(e+f*x)))-3/4*a*cos(e+f*x)/((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)*(c+d*sin(e+f*x))^3,x,6,4/105*(c-17*d)*d*(c+d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f+4/315*a^2*(c-17*d)*(c+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^2*(c-17*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d*f*sqrt(a+a*sin(e+f*x)))-2/9*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d*f*sqrt(a+a*sin(e+f*x)))+8/315*a*(c-17*d)*(5*c-d)*(c+d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^2,x,4,-4/35*(7*c-d)*d*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-2/7*d^2*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(a*f)-8/105*a^2*(35*c^2+42*c*d+19*d^2)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/105*a*(35*c^2+42*c*d+19*d^2)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x)),x,3,-2/5*d*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-8/15*a^2*(5*c+3*d)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/15*a*(5*c+3*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2),x,2,-8/3*a^2*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/3*a*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x)),x,4,2*a^(3/2)*(c-d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(3/2)*f*sqrt(c+d))-2*a^2*cos(e+f*x)/(d*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^2,x,4,-a^(3/2)*(c+3*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(3/2)*(c+d)^(3/2)*f)+a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^3,x,5,-1/4*a^(3/2)*(c+7*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(d^(3/2)*(c+d)^(5/2)*f)+1/2*a^2*(c-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^2*(c+7*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))^(5/2)*(c+d*sin(e+f*x))^3,x,6,-4/1155*a*(c+d)*(3*c^2-38*c*d+355*d^2)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-4/3465*a^3*(c+d)*(15*c^2+10*c*d+7*d^2)*(3*c^2-38*c*d+355*d^2)*cos(e+f*x)/(d^2*f*sqrt(a+a*sin(e+f*x)))-2/693*a^3*(3*c^2-38*c*d+355*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^3*(3*c-23*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^2*(5*c-d)*(c+d)*(3*c^2-38*c*d+355*d^2)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*f)-2/11*a^2*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))^(5/2)*(c+d*sin(e+f*x))^2,x,5,-2/105*a*(21*c^2+30*c*d+13*d^2)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-4/63*(9*c-d)*d*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/f-2/9*d^2*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*f)-64/315*a^3*(21*c^2+30*c*d+13*d^2)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-16/315*a^2*(21*c^2+30*c*d+13*d^2)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x)),x,4,-2/35*a*(7*c+5*d)*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-2/7*d*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/f-64/105*a^3*(7*c+5*d)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-16/105*a^2*(7*c+5*d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(5/2),x,3,-2/5*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/f-64/15*a^3*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-16/15*a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/f],
[(a+a*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x)),x,4,-2*a^(5/2)*(c-d)^2*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^3*(3*c-7*d)*cos(e+f*x)/(d^2*f*sqrt(a+a*sin(e+f*x)))-2/3*a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*f)],
[(a+a*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^2,x,4,a^(5/2)*(c-d)*(3*c+5*d)*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^3*(3*c+d)*cos(e+f*x)/(d^2*(c+d)*f*sqrt(a+a*sin(e+f*x)))+a^2*(c-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))^(5/2)/(c+d*sin(e+f*x))^3,x,4,-1/4*a^(5/2)*(3*c^2+10*c*d+19*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)+3/4*a^3*(c-d)*(c+3*d)*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^2*(c-d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^2)],

# m<0
[(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^(1/2),x,6,-(c-d)^3*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-4/15*d*(21*c^2-12*c*d+7*d^2)*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/5*d*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*sqrt(a+a*sin(e+f*x)))-2/15*(9*c-d)*d^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a*f)],
[(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^(1/2),x,4,-(c-d)^2*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-4/3*(3*c-d)*d*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))-2/3*d^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a*f)],
[(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^(1/2),x,3,-(c-d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-2*d*cos(e+f*x)/(f*sqrt(a+a*sin(e+f*x)))],
[1/(a+a*sin(e+f*x))^(1/2),x,2,-arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))],
[1/((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))),x,5,-arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/((c-d)*f*sqrt(a))+2*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(d)/((c-d)*f*sqrt(a)*sqrt(c+d))],
[1/((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^2),x,6,-arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/((c-d)^2*f*sqrt(a))+(3*c+d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(d)/((c-d)^2*(c+d)^(3/2)*f*sqrt(a))+d*cos(e+f*x)/((c^2-d^2)*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^3),x,7,-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*(15*c^2+10*c*d+7*d^2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(d)/((c-d)^3*(c+d)^(5/2)*f*sqrt(a))+1/2*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*d*(7*c+d)*cos(e+f*x)/((c^2-d^2)^2*f*(c+d*sin(e+f*x))*sqrt(a+a*sin(e+f*x)))],
[(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^(3/2),x,6,-1/2*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(c-d)^2*(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))+1/3*d*(3*c^2-24*c*d+13*d^2)*cos(e+f*x)/(a*f*sqrt(a+a*sin(e+f*x)))+1/6*(3*c-7*d)*d^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(a^2*f)],
[(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^(3/2),x,4,-1/2*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(c-d)*(c+7*d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*f*sqrt(2))+1/2*(c-5*d)*d*cos(e+f*x)/(a*f*sqrt(a+a*sin(e+f*x)))],
[(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^(3/2),x,3,-1/2*(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(c+3*d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*f*sqrt(2))],
[1/(a+a*sin(e+f*x))^(3/2),x,3,-1/2*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(3/2))-1/2*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*f*sqrt(2))],
[1/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))),x,6,-1/2*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(3/2))-1/2*(c-5*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*d^(3/2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*(c-d)^2*f*sqrt(c+d))],
[1/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^2),x,7,-d^(3/2)*(5*c+3*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*(c-d)^3*(c+d)^(3/2)*f)-1/2*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x)))-1/2*(c-9*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))-1/2*d*(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)))],
[1/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^3),x,8,-1/4*d^(3/2)*(35*c^2+42*c*d+19*d^2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(a^(3/2)*(c-d)^4*(c+d)^(5/2)*f)-1/2*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^2)-1/2*(c-13*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))-1/2*d*(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*(2*c+d)*(c+7*d)*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)))],
[(c+d*sin(e+f*x))^3/(a+a*sin(e+f*x))^(5/2),x,6,-1/16*(c-d)^2*(3*c+13*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/4*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(f*(a+a*sin(e+f*x))^(5/2))-3/16*(c-d)*(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))+1/4*(c-9*d)*d^2*cos(e+f*x)/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[(c+d*sin(e+f*x))^2/(a+a*sin(e+f*x))^(5/2),x,4,-3/16*(c-d)*(c+3*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/4*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(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))],
[(c+d*sin(e+f*x))/(a+a*sin(e+f*x))^(5/2),x,4,-1/4*(c-d)*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(3*c+5*d)*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2))-1/16*(3*c+5*d)*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*f*sqrt(2))],
[1/(a+a*sin(e+f*x))^(5/2),x,4,-1/4*cos(e+f*x)/(f*(a+a*sin(e+f*x))^(5/2))-3/16*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2))-3/16*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*f*sqrt(2))],
[1/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))),x,7,-1/4*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(5/2))-1/16*(3*c-11*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^(3/2))-1/16*(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^(5/2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))/(a^(5/2)*(c-d)^3*f*sqrt(c+d))],
[1/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^2),x,8,d^(5/2)*(7*c+5*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)^4*(c+d)^(3/2)*f)-1/4*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x)))-3/16*(c-5*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*(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*(c-7*d)*d*(3*c+5*d)*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)))],
[1/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^3),x,9,3/4*d^(5/2)*(21*c^2+30*c*d+13*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)^5*(c+d)^(5/2)*f)-1/4*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*c-19*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)-3/16*(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*(3*c^2-20*c*d-31*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)))-3/16*d*(c+3*d)*(c^2-10*c*d-7*d^2)*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/2) (c+d Sin[e+f x])^(n/2)

# m>0
[(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(5/2),x,5,-5/8*(c+d)^3*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(a)/(f*sqrt(d))-5/12*a*(c+d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))-1/3*a*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(f*sqrt(a+a*sin(e+f*x)))-5/8*a*(c+d)^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(3/2),x,4,-3/4*(c+d)^2*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(a)/(f*sqrt(d))-1/2*a*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))-3/4*a*(c+d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(1/2),x,3,-(c+d)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(a)/(f*sqrt(d))-a*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^(1/2),x,2,-2*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(a)/(f*sqrt(d))],
[(a+a*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^(3/2),x,1,-2*a*cos(e+f*x)/((c+d)*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^(5/2),x,2,-2/3*a*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-4/3*a*cos(e+f*x)/((c+d)^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^(7/2),x,3,-2/5*a*cos(e+f*x)/((c+d)*f*(c+d*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-8/15*a*cos(e+f*x)/((c+d)^2*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-16/15*a*cos(e+f*x)/((c+d)^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(5/2),x,7,5/64*a^(3/2)*(c-15*d)*(c+d)^3*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(3/2)*f)+5/96*a^2*(c-15*d)*(c+d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f*sqrt(a+a*sin(e+f*x)))+1/24*a^2*(c-15*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d*f*sqrt(a+a*sin(e+f*x)))-1/4*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(7/2)/(d*f*sqrt(a+a*sin(e+f*x)))+5/64*a^2*(c-15*d)*(c+d)^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(3/2),x,6,1/8*a^(3/2)*(c-11*d)*(c+d)^2*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(3/2)*f)+1/12*a^2*(c-11*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f*sqrt(a+a*sin(e+f*x)))-1/3*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d*f*sqrt(a+a*sin(e+f*x)))+1/8*a^2*(c-11*d)*(c+d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(1/2),x,5,1/4*a^(3/2)*(c-7*d)*(c+d)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(3/2)*f)-1/2*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f*sqrt(a+a*sin(e+f*x)))+1/4*a^2*(c-7*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^(1/2),x,4,a^(3/2)*(c-3*d)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(3/2)*f)-a^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^(3/2),x,4,-2*a^(3/2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(3/2)*f)+2*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^(5/2),x,3,2/3*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-2/3*a^2*(c+5*d)*cos(e+f*x)/(d*(c+d)^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^(7/2),x,4,2/5*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-2/15*a^2*(c+9*d)*cos(e+f*x)/(d*(c+d)^2*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-4/15*a^2*(c+9*d)*cos(e+f*x)/(d*(c+d)^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^(9/2),x,5,2/7*a^2*(c-d)*cos(e+f*x)/(d*(c+d)*f*(c+d*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))-2/35*a^2*(c+13*d)*cos(e+f*x)/(d*(c+d)^2*f*(c+d*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-8/105*a^2*(c+13*d)*cos(e+f*x)/(d*(c+d)^3*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-16/105*a^2*(c+13*d)*cos(e+f*x)/(d*(c+d)^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(5/2),x,7,-1/128*a^(5/2)*(c+d)^3*(3*c^2-34*c*d+283*d^2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(5/2)*f)-1/192*a^3*(c+d)*(3*c^2-34*c*d+283*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f*sqrt(a+a*sin(e+f*x)))-1/240*a^3*(3*c^2-34*c*d+283*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d^2*f*sqrt(a+a*sin(e+f*x)))+3/40*a^3*(c-7*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(7/2)/(d^2*f*sqrt(a+a*sin(e+f*x)))-1/5*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x))/(d*f)-1/128*a^3*(c+d)^2*(3*c^2-34*c*d+283*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(3/2),x,6,-1/64*a^(5/2)*(c+d)^2*(3*c^2-26*c*d+163*d^2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(5/2)*f)-1/96*a^3*(3*c^2-26*c*d+163*d^2)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f*sqrt(a+a*sin(e+f*x)))+1/24*a^3*(3*c-17*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d^2*f*sqrt(a+a*sin(e+f*x)))-1/4*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(d*f)-1/64*a^3*(c+d)*(3*c^2-26*c*d+163*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(1/2),x,5,-1/8*a^(5/2)*(c+d)*(c^2-6*c*d+25*d^2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(5/2)*f)+1/12*a^3*(3*c-13*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f*sqrt(a+a*sin(e+f*x)))-1/3*a^2*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/(d*f)-1/8*a^3*(c^2-6*c*d+25*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(1/2),x,4,-1/4*a^(5/2)*(3*c^2-10*c*d+19*d^2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(5/2)*f)+3/4*a^3*(c-3*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt(a+a*sin(e+f*x)))-1/2*a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(d*f)],
[(a+a*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(3/2),x,4,a^(5/2)*(3*c-5*d)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(5/2)*f)+2*a^2*(c-d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*(c+d)*f*sqrt(c+d*sin(e+f*x)))-a^3*(3*c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*(c+d)*f*sqrt(a+a*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(5/2),x,4,-2*a^(5/2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(d^(5/2)*f)+2/3*a^2*(c-d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^(3/2))+2/3*a^3*(c-d)*(3*c+7*d)*cos(e+f*x)/(d^2*(c+d)^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(7/2),x,3,2/15*a^3*(c-d)*(3*c+11*d)*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+2/5*a^2*(c-d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^(5/2))-2/15*a^3*(3*c^2+14*c*d+43*d^2)*cos(e+f*x)/(d^2*(c+d)^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(9/2),x,4,6/35*a^3*(c-d)*(c+5*d)*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-2/105*a^3*(3*c^2+22*c*d+115*d^2)*cos(e+f*x)/(d^2*(c+d)^3*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+2/7*a^2*(c-d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^(7/2))-4/105*a^3*(3*c^2+22*c*d+115*d^2)*cos(e+f*x)/(d^2*(c+d)^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(a+a*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(11/2),x,5,2/63*a^3*(c-d)*(3*c+19*d)*cos(e+f*x)/(d^2*(c+d)^2*f*(c+d*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))-2/105*a^3*(c^2+10*c*d+73*d^2)*cos(e+f*x)/(d^2*(c+d)^3*f*(c+d*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-8/315*a^3*(c^2+10*c*d+73*d^2)*cos(e+f*x)/(d^2*(c+d)^4*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+2/9*a^2*(c-d)*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(d*(c+d)*f*(c+d*sin(e+f*x))^(9/2))-16/315*a^3*(c^2+10*c*d+73*d^2)*cos(e+f*x)/(d^2*(c+d)^5*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],

# m<0
[(c+d*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(1/2),x,7,-(c-d)^(5/2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-1/4*(15*c^2-10*c*d+7*d^2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(d)/(f*sqrt(a))-1/2*d*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*sqrt(a+a*sin(e+f*x)))-1/4*(7*c-d)*d*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(1/2),x,6,-(c-d)^(3/2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(2)/(f*sqrt(a))-(3*c-d)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(d)/(f*sqrt(a))-d*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+a*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^(1/2),x,5,-arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(2)*sqrt(c-d)/(f*sqrt(a))-2*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(d)/(f*sqrt(a))],
[1/((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(1/2)),x,2,-arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(2)/(f*sqrt(a)*sqrt(c-d))],
[1/((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(3/2)),x,4,-arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(2)/((c-d)^(3/2)*f*sqrt(a))+2*d*cos(e+f*x)/((c^2-d^2)*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(5/2)),x,5,-arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(2)/((c-d)^(5/2)*f*sqrt(a))+2/3*d*cos(e+f*x)/((c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+2/3*d*(5*c+d)*cos(e+f*x)/((c^2-d^2)^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(3/2),x,7,-(5*c-3*d)*d^(3/2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(3/2)*f)-1/2*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^(3/2))-1/2*(c-d)^(3/2)*(c+9*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(3/2)*f*sqrt(2))+1/2*(c-3*d)*d*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(3/2),x,6,-2*d^(3/2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(3/2)*f)-1/2*(c+5*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(c-d)/(a^(3/2)*f*sqrt(2))-1/2*(c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^(3/2))],
[(c+d*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^(3/2),x,4,-1/2*(c+d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(3/2)*f*sqrt(2)*sqrt(c-d))-1/2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^(3/2))],
[1/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(1/2)),x,4,-1/2*(c-3*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(3/2)*(c-d)^(3/2)*f*sqrt(2))-1/2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((c-d)*f*(a+a*sin(e+f*x))^(3/2))],
[1/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(3/2)),x,5,-1/2*(c-7*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(3/2)*(c-d)^(5/2)*f*sqrt(2))-1/2*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(3/2)*sqrt(c+d*sin(e+f*x)))-1/2*d*(c+5*d)*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(5/2)),x,6,-1/2*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(3/2))-1/2*(c-11*d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(3/2)*(c-d)^(7/2)*f*sqrt(2))-1/6*d*(3*c+7*d)*cos(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-1/6*d*(3*c^2+38*c*d+19*d^2)*cos(e+f*x)/(a*(c-d)^3*(c+d)^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(5/2),x,7,-2*d^(5/2)*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(5/2)*f)-1/4*(c-d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(3*c^2+14*c*d+43*d^2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(c-d)/(a^(5/2)*f*sqrt(2))-1/16*(c-d)*(3*c+11*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*f*(a+a*sin(e+f*x))^(3/2))],
[(c+d*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(5/2),x,5,-3/16*(c+d)^2*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(5/2)*f*sqrt(2)*sqrt(c-d))-1/4*(c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(3*c+7*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*f*(a+a*sin(e+f*x))^(3/2))],
[(c+d*sin(e+f*x))^(1/2)/(a+a*sin(e+f*x))^(5/2),x,5,-1/16*(3*c-5*d)*(c+d)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(5/2)*(c-d)^(3/2)*f*sqrt(2))-1/4*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*(a+a*sin(e+f*x))^(5/2))-1/16*(3*c-d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*(c-d)*f*(a+a*sin(e+f*x))^(3/2))],
[1/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(1/2)),x,5,-1/16*(3*c^2-10*c*d+19*d^2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(5/2)*(c-d)^(5/2)*f*sqrt(2))-1/4*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((c-d)*f*(a+a*sin(e+f*x))^(5/2))-3/16*(c-3*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(a*(c-d)^2*f*(a+a*sin(e+f*x))^(3/2))],
[1/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(3/2)),x,6,-3/16*(c^2-6*c*d+25*d^2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(5/2)*(c-d)^(7/2)*f*sqrt(2))-1/4*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(5/2)*sqrt(c+d*sin(e+f*x)))-1/16*(3*c-13*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^(3/2)*sqrt(c+d*sin(e+f*x)))-1/16*(c-7*d)*d*(3*c+7*d)*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[1/((a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(5/2)),x,7,-1/4*cos(e+f*x)/((c-d)*f*(a+a*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(3/2))-1/16*(3*c-17*d)*cos(e+f*x)/(a*(c-d)^2*f*(a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(3/2))-1/16*(3*c^2-26*c*d+163*d^2)*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(5/2)*(c-d)^(9/2)*f*sqrt(2))-1/48*d*(9*c^2-54*c*d-95*d^2)*cos(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-1/48*d*(9*c^3-57*c^2*d-493*c*d^2-299*d^3)*cos(e+f*x)/(a^2*(c-d)^4*(c+d)^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],

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

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

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

# m>0
[(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^3,x,3,1/8*(8*a*c^3+12*b*c^2*d+12*a*c*d^2+3*b*d^3)*x-1/6*(4*a*d*(4*c^2+d^2)+3*b*(c^3+4*c*d^2))*cos(e+f*x)/f-1/24*d*(6*b*c^2+20*a*c*d+9*b*d^2)*cos(e+f*x)*sin(e+f*x)/f-1/12*(3*b*c+4*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/f-1/4*b*cos(e+f*x)*(c+d*sin(e+f*x))^3/f],
[(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^2,x,2,1/2*(2*b*c*d+a*(2*c^2+d^2))*x-2/3*(3*a*c*d+b*(c^2+d^2))*cos(e+f*x)/f-1/6*d*(2*b*c+3*a*d)*cos(e+f*x)*sin(e+f*x)/f-1/3*b*cos(e+f*x)*(c+d*sin(e+f*x))^2/f],
[(a+b*sin(e+f*x))*(c+d*sin(e+f*x)),x,1,1/2*(2*a*c+b*d)*x-(b*c+a*d)*cos(e+f*x)/f-1/2*b*d*cos(e+f*x)*sin(e+f*x)/f],
[a+b*sin(e+f*x),x,2,a*x-b*cos(e+f*x)/f],
[(a+b*sin(e+f*x))/(c+d*sin(e+f*x)),x,4,b*x/d-2*(b*c-a*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d*f*sqrt(c^2-d^2))],
[(a+b*sin(e+f*x))/(c+d*sin(e+f*x))^2,x,5,2*(a*c-b*d)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c^2-d^2)^(3/2)*f)-(b*c-a*d)*cos(e+f*x)/((c^2-d^2)*f*(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))/(c+d*sin(e+f*x))^3,x,6,-(3*b*c*d-a*(2*c^2+d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c^2-d^2)^(5/2)*f)-1/2*(b*c-a*d)*cos(e+f*x)/((c^2-d^2)*f*(c+d*sin(e+f*x))^2)+1/2*(3*a*c*d-b*(c^2+2*d^2))*cos(e+f*x)/((c^2-d^2)^2*f*(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^3,x,4,1/8*(6*a*b*d*(4*c^2+d^2)+b^2*c*(4*c^2+9*d^2)+4*a^2*(2*c^3+3*c*d^2))*x-1/30*(20*a^2*d^2*(4*c^2+d^2)+30*a*b*c*d*(c^2+4*d^2)-b^2*(3*c^4-52*c^2*d^2-16*d^4))*cos(e+f*x)/(d*f)-1/120*(100*a^2*c*d^2+30*a*b*d*(2*c^2+3*d^2)-b^2*(6*c^3-71*c*d^2))*cos(e+f*x)*sin(e+f*x)/f-1/60*(4*(5*a^2+4*b^2)*d^2-3*b*c*(b*c-10*a*d))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d*f)+1/20*b*(b*c-10*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d*f)-1/5*b^2*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d*f)],
[(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^2,x,3,1/8*(16*a*b*c*d+4*a^2*(2*c^2+d^2)+b^2*(4*c^2+3*d^2))*x-1/6*(8*a^2*b*c*d+8*b^3*c*d-a^3*d^2+4*a*b^2*(3*c^2+2*d^2))*cos(e+f*x)/(b*f)-1/24*(2*a*d*(8*b*c-a*d)+3*b^2*(4*c^2+3*d^2))*cos(e+f*x)*sin(e+f*x)/f-1/12*d*(8*b*c-a*d)*cos(e+f*x)*(a+b*sin(e+f*x))^2/(b*f)-1/4*d^2*cos(e+f*x)*(a+b*sin(e+f*x))^3/(b*f)],
[(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x)),x,2,1/2*(2*a^2*c+b^2*c+2*a*b*d)*x-2/3*(3*a*b*c+a^2*d+b^2*d)*cos(e+f*x)/f-1/6*b*(3*b*c+2*a*d)*cos(e+f*x)*sin(e+f*x)/f-1/3*d*cos(e+f*x)*(a+b*sin(e+f*x))^2/f],
[(a+b*sin(e+f*x))^2,x,1,1/2*(2*a^2+b^2)*x-2*a*b*cos(e+f*x)/f-1/2*b^2*cos(e+f*x)*sin(e+f*x)/f],
[(a+b*sin(e+f*x))^2/(c+d*sin(e+f*x)),x,5,-b*(b*c-2*a*d)*x/d^2-b^2*cos(e+f*x)/(d*f)+2*(b*c-a*d)^2*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^2*f*sqrt(c^2-d^2))],
[(a+b*sin(e+f*x))^2/(c+d*sin(e+f*x))^2,x,5,b^2*x/d^2-2*(b*c-a*d)*(a*c*d+b*(c^2-2*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)+(b*c-a*d)^2*cos(e+f*x)/(d*(c^2-d^2)*f*(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2/(c+d*sin(e+f*x))^3,x,6,-(6*a*b*c*d-a^2*(2*c^2+d^2)-b^2*(c^2+2*d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c^2-d^2)^(5/2)*f)+1/2*(b*c-a*d)^2*cos(e+f*x)/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^2)-1/2*(b*c-a*d)*(3*a*c*d+b*(c^2-4*d^2))*cos(e+f*x)/(d*(c^2-d^2)^2*f*(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2/(c+d*sin(e+f*x))^4,x,7,-(2*a*b*d*(4*c^2+d^2)-b^2*c*(c^2+4*d^2)-a^2*(2*c^3+3*c*d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c^2-d^2)^(7/2)*f)+1/3*(b*c-a*d)^2*cos(e+f*x)/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^3)-1/6*(b*c-a*d)*(5*a*c*d+b*(c^2-6*d^2))*cos(e+f*x)/(d*(c^2-d^2)^2*f*(c+d*sin(e+f*x))^2)+1/6*(a^2*d^2*(11*c^2+4*d^2)-a*b*(4*c^3*d+26*c*d^3)-b^2*(c^4-10*c^2*d^2-6*d^4))*cos(e+f*x)/(d*(c^2-d^2)^3*f*(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^3,x,5,1/16*(18*a^2*b*d*(4*c^2+d^2)+b^3*d*(18*c^2+5*d^2)+6*a*b^2*c*(4*c^2+9*d^2)+8*a^3*(2*c^3+3*c*d^2))*x-(3*a*b^2*d*(3*c^2+d^2)+3*a^2*b*c*(c^2+3*d^2)+b^3*c*(c^2+3*d^2)+a^3*(3*c^2*d+d^3))*cos(e+f*x)/f+1/3*(b*c+a*d)*(8*a*b*c*d+a^2*d^2+b^2*(c^2+6*d^2))*cos(e+f*x)^3/f-3/5*b^2*d^2*(b*c+a*d)*cos(e+f*x)^5/f-1/16*(24*a^3*c*d^2+18*a^2*b*d*(4*c^2+d^2)+b^3*d*(18*c^2+5*d^2)+6*a*b^2*c*(4*c^2+9*d^2))*cos(e+f*x)*sin(e+f*x)/f-5/24*b^3*d^3*cos(e+f*x)*sin(e+f*x)^3/f-3/4*b*d*(b^2*c^2+3*a*b*c*d+a^2*d^2)*cos(e+f*x)*sin(e+f*x)^3/f-1/6*b^3*d^3*cos(e+f*x)*sin(e+f*x)^5/f,1/16*(18*a^2*b*d*(4*c^2+d^2)+b^3*d*(18*c^2+5*d^2)+6*a*b^2*c*(4*c^2+9*d^2)+8*a^3*(2*c^3+3*c*d^2))*x-1/60*(40*a^3*d^3*(4*c^2+d^2)+90*a^2*b*c*d^2*(c^2+4*d^2)-6*a*b^2*d*(3*c^4-52*c^2*d^2-16*d^4)+b^3*(2*c^5+17*c^3*d^2+96*c*d^4))*cos(e+f*x)/(d^2*f)-1/240*(200*a^3*c*d^3+90*a^2*b*d^2*(2*c^2+3*d^2)-6*a*b^2*d*(6*c^3-71*c*d^2)+b^3*(4*c^4+36*c^2*d^2+75*d^4))*cos(e+f*x)*sin(e+f*x)/(d*f)-1/120*(90*a^2*b*c*d^2+40*a^3*d^3+b^3*(2*c^3+21*c*d^2)-a*b^2*(18*c^2*d-96*d^3))*cos(e+f*x)*(c+d*sin(e+f*x))^2/(d^2*f)+1/120*b*(18*a*b*c*d-90*a^2*d^2-b^2*(2*c^2+25*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^3/(d^2*f)+1/30*b^2*(2*b*c-13*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^4/(d^2*f)-1/6*b^2*cos(e+f*x)*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^4/(d*f)],
[(a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^2,x,4,1/8*(24*a^2*b*c*d+6*b^3*c*d+4*a^3*(2*c^2+d^2)+3*a*b^2*(4*c^2+3*d^2))*x-1/30*(30*a^3*b*c*d+120*a*b^3*c*d-3*a^4*d^2+4*b^4*(5*c^2+4*d^2)+4*a^2*b^2*(20*c^2+13*d^2))*cos(e+f*x)/(b*f)-1/120*(60*a^2*b*c*d+90*b^3*c*d-6*a^3*d^2+a*b^2*(100*c^2+71*d^2))*cos(e+f*x)*sin(e+f*x)/f-1/60*(3*a*d*(10*b*c-a*d)+4*b^2*(5*c^2+4*d^2))*cos(e+f*x)*(a+b*sin(e+f*x))^2/(b*f)-1/20*d*(10*b*c-a*d)*cos(e+f*x)*(a+b*sin(e+f*x))^3/(b*f)-1/5*d^2*cos(e+f*x)*(a+b*sin(e+f*x))^4/(b*f)],
[(a+b*sin(e+f*x))^3*(c+d*sin(e+f*x)),x,3,1/8*(8*a^3*c+12*a*b^2*c+12*a^2*b*d+3*b^3*d)*x-1/6*(16*a^2*b*c+4*b^3*c+3*a^3*d+12*a*b^2*d)*cos(e+f*x)/f-1/24*b*(20*a*b*c+6*a^2*d+9*b^2*d)*cos(e+f*x)*sin(e+f*x)/f-1/12*(4*b*c+3*a*d)*cos(e+f*x)*(a+b*sin(e+f*x))^2/f-1/4*d*cos(e+f*x)*(a+b*sin(e+f*x))^3/f],
[(a+b*sin(e+f*x))^3,x,2,1/2*a*(2*a^2+3*b^2)*x-2/3*b*(4*a^2+b^2)*cos(e+f*x)/f-5/6*a*b^2*cos(e+f*x)*sin(e+f*x)/f-1/3*b*cos(e+f*x)*(a+b*sin(e+f*x))^2/f],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x)),x,6,-1/2*b*(6*a*b*c*d-6*a^2*d^2-b^2*(2*c^2+d^2))*x/d^3+1/2*b^2*(2*b*c-5*a*d)*cos(e+f*x)/(d^2*f)-1/2*b^2*cos(e+f*x)*(a+b*sin(e+f*x))/(d*f)-2*(b*c-a*d)^3*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^3*f*sqrt(c^2-d^2))],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x))^2,x,6,-b^2*(2*b*c-3*a*d)*x/d^3+2*(b*c-a*d)^2*(2*b*c^2+a*c*d-3*b*d^2)*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*a*b*c*d-a^2*d^2-b^2*(2*c^2-d^2))*cos(e+f*x)/(d^2*(c^2-d^2)*f)+(b*c-a*d)^2*cos(e+f*x)*(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x))^3,x,6,b^3*x/d^3-(9*a^2*b*c*d^4-a^3*d^3*(2*c^2+d^2)-3*a*b^2*d^3*(c^2+2*d^2)+b^3*(2*c^5-5*c^3*d^2+6*c*d^4))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/(d^3*(c^2-d^2)^(5/2)*f)+1/2*(b*c-a*d)^2*cos(e+f*x)*(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^2)+1/2*(b*c-a*d)^2*(2*b*c^2+3*a*c*d-5*b*d^2)*cos(e+f*x)/(d^2*(c^2-d^2)^2*f*(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x))^4,x,7,-(a*c-b*d)*(10*a*b*c*d-b^2*(3*c^2+2*d^2)-a^2*(2*c^2+3*d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((c^2-d^2)^(7/2)*f)+1/3*(b*c-a*d)^2*cos(e+f*x)*(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^3)+1/6*(b*c-a*d)^2*(2*b*c^2+5*a*c*d-7*b*d^2)*cos(e+f*x)/(d^2*(c^2-d^2)^2*f*(c+d*sin(e+f*x))^2)-1/6*(b*c-a*d)*(5*a*b*c*d*(c^2-7*d^2)+a^2*d^2*(11*c^2+4*d^2)+b^2*(2*c^4-5*c^2*d^2+18*d^4))*cos(e+f*x)/(d^2*(c^2-d^2)^3*f*(c+d*sin(e+f*x)))],
[(b*B/a+B*sin(x))/(a+b*sin(x)),x,4,B*x/b-2*B*arctan((b+a*tan(1/2*x))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a*b)],
[(a*B/b+B*sin(x))/(a+b*sin(x)),x,2,B*x/b],
[(a+b*sin(x))/(b+a*sin(x))^2,x,2,-cos(x)/(b+a*sin(x))],
[(2-sin(x))/(2+sin(x)),x,2,-x+4*x/sqrt(3)+8*arctan(cos(x)/(2+sin(x)+sqrt(3)))/sqrt(3)],

# m<0
[(c+d*sin(e+f*x))^4/(a+b*sin(e+f*x)),x,7,1/2*d*(8*a^2*b*c*d^2-2*a^3*d^3+4*b^3*c*(2*c^2+d^2)-a*b^2*d*(12*c^2+d^2))*x/b^4+1/3*d^2*(12*a*b*c*d-3*a^2*d^2-b^2*(17*c^2+2*d^2))*cos(e+f*x)/(b^3*f)-1/6*d^3*(8*b*c-3*a*d)*cos(e+f*x)*sin(e+f*x)/(b^2*f)-1/3*d^2*cos(e+f*x)*(c+d*sin(e+f*x))^2/(b*f)+2*(b*c-a*d)^4*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^4*f*sqrt(a^2-b^2))],
[(c+d*sin(e+f*x))^3/(a+b*sin(e+f*x)),x,6,-1/2*d*(6*a*b*c*d-2*a^2*d^2-b^2*(6*c^2+d^2))*x/b^3-1/2*d^2*(5*b*c-2*a*d)*cos(e+f*x)/(b^2*f)-1/2*d^2*cos(e+f*x)*(c+d*sin(e+f*x))/(b*f)+2*(b*c-a*d)^3*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^3*f*sqrt(a^2-b^2))],
[(c+d*sin(e+f*x))^2/(a+b*sin(e+f*x)),x,5,d*(2*b*c-a*d)*x/b^2-d^2*cos(e+f*x)/(b*f)+2*(b*c-a*d)^2*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^2*f*sqrt(a^2-b^2))],
[(c+d*sin(e+f*x))/(a+b*sin(e+f*x)),x,4,d*x/b+2*(b*c-a*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b*f*sqrt(a^2-b^2))],
[1/(a+b*sin(e+f*x)),x,3,2*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(f*sqrt(a^2-b^2))],
[1/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))),x,7,2*b*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((b*c-a*d)*f*sqrt(a^2-b^2))-2*d*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)*f*sqrt(c^2-d^2))],
[1/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))^2),x,8,2*d*(a*c*d-b*(2*c^2-d^2))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)^2*(c^2-d^2)^(3/2)*f)-d^2*cos(e+f*x)/((b*c-a*d)*(c^2-d^2)*f*(c+d*sin(e+f*x)))+2*b^2*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((b*c-a*d)^2*f*sqrt(a^2-b^2))],
[1/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))^3),x,9,d*(6*a*b*c^3*d-a^2*d^2*(2*c^2+d^2)-b^2*(6*c^4-5*c^2*d^2+2*d^4))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)^3*(c^2-d^2)^(5/2)*f)-1/2*d^2*cos(e+f*x)/((b*c-a*d)*(c^2-d^2)*f*(c+d*sin(e+f*x))^2)-1/2*d^2*(5*b*c^2-3*a*c*d-2*b*d^2)*cos(e+f*x)/((b*c-a*d)^2*(c^2-d^2)^2*f*(c+d*sin(e+f*x)))+2*b^3*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((b*c-a*d)^3*f*sqrt(a^2-b^2))],
[(c+d*sin(e+f*x))^4/(a+b*sin(e+f*x))^2,x,7,-1/2*d^2*(16*a*b*c*d-6*a^2*d^2-b^2*(12*c^2+d^2))*x/b^4+2*(b*c-a*d)^3*(a*b*c+3*a^2*d-4*b^2*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^4*(a^2-b^2)^(3/2)*f)+d*(2*b*c-a*d)*(2*a*b*c*d-3*a^2*d^2-b^2*(c^2-2*d^2))*cos(e+f*x)/(b^3*(a^2-b^2)*f)+1/2*d^2*(4*a*b*c*d-3*a^2*d^2-b^2*(2*c^2-d^2))*cos(e+f*x)*sin(e+f*x)/(b^2*(a^2-b^2)*f)+(b*c-a*d)^2*cos(e+f*x)*(c+d*sin(e+f*x))^2/(b*(a^2-b^2)*f*(a+b*sin(e+f*x)))],
[(c+d*sin(e+f*x))^3/(a+b*sin(e+f*x))^2,x,6,d^2*(3*b*c-2*a*d)*x/b^3+2*(b*c-a*d)^2*(a*b*c+2*a^2*d-3*b^2*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^3*(a^2-b^2)^(3/2)*f)+d*(2*a*b*c*d-2*a^2*d^2-b^2*(c^2-d^2))*cos(e+f*x)/(b^2*(a^2-b^2)*f)+(b*c-a*d)^2*cos(e+f*x)*(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*(a+b*sin(e+f*x)))],
[(c+d*sin(e+f*x))^2/(a+b*sin(e+f*x))^2,x,5,d^2*x/b^2+2*(b*c-a*d)*(a*b*c+a^2*d-2*b^2*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^2*(a^2-b^2)^(3/2)*f)+(b*c-a*d)^2*cos(e+f*x)/(b*(a^2-b^2)*f*(a+b*sin(e+f*x)))],
[(c+d*sin(e+f*x))/(a+b*sin(e+f*x))^2,x,5,2*(a*c-b*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*f)+(b*c-a*d)*cos(e+f*x)/((a^2-b^2)*f*(a+b*sin(e+f*x)))],
[1/(a+b*sin(e+f*x))^2,x,5,2*a*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*f)+b*cos(e+f*x)/((a^2-b^2)*f*(a+b*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))),x,8,2*b*(a*b*c-2*a^2*d+b^2*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*(b*c-a*d)^2*f)+b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x)))+2*d^2*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)^2*f*sqrt(c^2-d^2))],
[1/((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^2),x,9,2*b^2*(a*b*c-3*a^2*d+2*b^2*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*(b*c-a*d)^3*f)+2*d^2*(3*b*c^2-a*c*d-2*b*d^2)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)^3*(c^2-d^2)^(3/2)*f)+d*(a^2*d^2+b^2*(c^2-2*d^2))*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)^2*(c^2-d^2)*f*(c+d*sin(e+f*x)))+b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))*(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^3),x,10,2*b^3*(a*b*c-4*a^2*d+3*b^2*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*(b*c-a*d)^4*f)-d^2*(2*a*b*c*d*(4*c^2-d^2)-a^2*d^2*(2*c^2+d^2)-3*b^2*(4*c^4-5*c^2*d^2+2*d^4))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)^4*(c^2-d^2)^(5/2)*f)+1/2*d*(a^2*d^2+b^2*(2*c^2-3*d^2))*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)^2*(c^2-d^2)*f*(c+d*sin(e+f*x))^2)+b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^2)-1/2*(3*a^3*c*d^4-3*a*b^2*c*d^4-a^2*b*d^3*(7*c^2-4*d^2)-b^3*(2*c^4*d-11*c^2*d^3+6*d^5))*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)^3*(c^2-d^2)^2*f*(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^5/(a+b*sin(e+f*x))^3,x,8,-1/2*d^3*(30*a*b*c*d-12*a^2*d^2-b^2*(20*c^2+d^2))*x/b^5+(b*c-a*d)^3*(6*a^3*b*c*d-12*a*b^3*c*d+12*a^4*d^2+a^2*b^2*(2*c^2-29*d^2)+b^4*(c^2+20*d^2))*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^5*(a^2-b^2)^(5/2)*f)-1/2*d*(30*a^4*b*c*d^3-12*a^5*d^4-a^3*b^2*d^2*(16*c^2-21*d^2)-b^5*c*d*(17*c^2-10*d^2)-a^2*b^3*c*d*(4*c^2+55*d^2)+a*b^4*(6*c^4+43*c^2*d^2-6*d^4))*cos(e+f*x)/(b^4*(a^2-b^2)^2*f)+1/2*d^2*(7*a^3*b*c*d^2-6*a^4*d^3+b^4*d*(8*c^2-d^2)+a^2*b^2*d*(c^2+10*d^2)-a*b^3*c*(3*c^2+16*d^2))*cos(e+f*x)*sin(e+f*x)/(b^3*(a^2-b^2)^2*f)+1/2*(b*c-a*d)^2*(3*a*b*c+4*a^2*d-7*b^2*d)*cos(e+f*x)*(c+d*sin(e+f*x))^2/(b^2*(a^2-b^2)^2*f*(a+b*sin(e+f*x)))+1/2*(b*c-a*d)^2*cos(e+f*x)*(c+d*sin(e+f*x))^3/(b*(a^2-b^2)*f*(a+b*sin(e+f*x))^2)],
[(c+d*sin(e+f*x))^4/(a+b*sin(e+f*x))^3,x,7,d^3*(4*b*c-3*a*d)*x/b^4+(b*c-a*d)^2*(4*a^3*b*c*d-10*a*b^3*c*d+6*a^4*d^2+a^2*b^2*(2*c^2-15*d^2)+b^4*(c^2+12*d^2))*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^4*(a^2-b^2)^(5/2)*f)+1/2*d^2*(2*a*b*c*d-3*a^2*d^2-b^2*(c^2-2*d^2))*cos(e+f*x)/(b^3*(a^2-b^2)*f)+3/2*(b*c-a*d)^3*(a*b*c+a^2*d-2*b^2*d)*cos(e+f*x)/(b^3*(a^2-b^2)^2*f*(a+b*sin(e+f*x)))+1/2*(b*c-a*d)^2*cos(e+f*x)*(c+d*sin(e+f*x))^2/(b*(a^2-b^2)*f*(a+b*sin(e+f*x))^2)],
[(c+d*sin(e+f*x))^3/(a+b*sin(e+f*x))^3,x,6,d^3*x/b^3+(b*c-a*d)*(2*a^3*b*c*d-8*a*b^3*c*d+2*a^4*d^2+a^2*b^2*(2*c^2-5*d^2)+b^4*(c^2+6*d^2))*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/(b^3*(a^2-b^2)^(5/2)*f)+1/2*(b*c-a*d)^2*(3*a*b*c+2*a^2*d-5*b^2*d)*cos(e+f*x)/(b^2*(a^2-b^2)^2*f*(a+b*sin(e+f*x)))+1/2*(b*c-a*d)^2*cos(e+f*x)*(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*(a+b*sin(e+f*x))^2)],
[(c+d*sin(e+f*x))^2/(a+b*sin(e+f*x))^3,x,6,-(6*a*b*c*d-a^2*(2*c^2+d^2)-b^2*(c^2+2*d^2))*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*f)+1/2*(b*c-a*d)^2*cos(e+f*x)/(b*(a^2-b^2)*f*(a+b*sin(e+f*x))^2)+1/2*(b*c-a*d)*(3*a*b*c+a^2*d-4*b^2*d)*cos(e+f*x)/(b*(a^2-b^2)^2*f*(a+b*sin(e+f*x)))],
[(c+d*sin(e+f*x))/(a+b*sin(e+f*x))^3,x,6,(2*a^2*c+b^2*c-3*a*b*d)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*f)+1/2*(b*c-a*d)*cos(e+f*x)/((a^2-b^2)*f*(a+b*sin(e+f*x))^2)+1/2*(3*a*b*c-a^2*d-2*b^2*d)*cos(e+f*x)/((a^2-b^2)^2*f*(a+b*sin(e+f*x)))],
[1/(a+b*sin(e+f*x))^3,x,6,(2*a^2+b^2)*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*f)+1/2*b*cos(e+f*x)/((a^2-b^2)*f*(a+b*sin(e+f*x))^2)+3/2*a*b*cos(e+f*x)/((a^2-b^2)^2*f*(a+b*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))),x,9,-b*(6*a^3*b*c*d-6*a^4*d^2-a^2*b^2*(2*c^2-5*d^2)-b^4*(c^2+2*d^2))*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*(b*c-a*d)^3*f)+1/2*b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))^2)+1/2*b^2*(3*a*b*c-5*a^2*d+2*b^2*d)*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^2*f*(a+b*sin(e+f*x)))-2*d^3*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)^3*f*sqrt(c^2-d^2))],
[1/((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^2),x,10,-b^2*(8*a^3*b*c*d-2*a*b^3*c*d-12*a^4*d^2-a^2*b^2*(2*c^2-15*d^2)-b^4*(c^2+6*d^2))*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*(b*c-a*d)^4*f)-2*d^3*(4*b*c^2-a*c*d-3*b*d^2)*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)^4*(c^2-d^2)^(3/2)*f)-1/2*d*(2*a^4*d^3+a^2*b^2*d*(7*c^2-11*d^2)-2*b^4*d*(2*c^2-3*d^2)-3*a*b^3*c*(c^2-d^2))*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^3*(c^2-d^2)*f*(c+d*sin(e+f*x)))+1/2*b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x)))+3/2*b^2*(a*b*c-2*a^2*d+b^2*d)*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^2*f*(a+b*sin(e+f*x))*(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^3),x,11,-b^3*(10*a^3*b*c*d-4*a*b^3*c*d-20*a^4*d^2-a^2*b^2*(2*c^2-29*d^2)-b^4*(c^2+12*d^2))*arctan((b+a*tan(1/2*(e+f*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*(b*c-a*d)^5*f)-d^3*(a^2*d^2*(2*c^2+d^2)-a*b*(10*c^3*d-4*c*d^3)+b^2*(20*c^4-29*c^2*d^2+12*d^4))*arctan((d+c*tan(1/2*(e+f*x)))/sqrt(c^2-d^2))/((b*c-a*d)^5*(c^2-d^2)^(5/2)*f)-1/2*d*(a^4*d^3-b^4*d*(5*c^2-6*d^2)+2*a^2*b^2*d*(4*c^2-5*d^2)-3*a*b^3*c*(c^2-d^2))*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^3*(c^2-d^2)*f*(c+d*sin(e+f*x))^2)+1/2*b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^2)+1/2*b^2*(3*a*b*c-7*a^2*d+4*b^2*d)*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^2*f*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^2)+3/2*d*(a^5*c*d^4-2*a^3*b^2*c*d^4+a*b^4*c*(c^4-2*c^2*d^2+2*d^4)+b^5*d*(2*c^4-7*c^2*d^2+4*d^4)-a^2*b^3*d*(3*c^4-12*c^2*d^2+7*d^4)-a^4*b*(3*c^2*d^3-2*d^5))*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^4*(c^2-d^2)^2*f*(c+d*sin(e+f*x)))],

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

# m>0
[(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(5/2),x,8,-2/35*(5*b*c+7*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/f-2/7*b*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/f-2/105*(15*b*c^2+56*a*c*d+25*b*d^2)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/f+2/105*(15*b*c^3+161*a*c^2*d+145*b*c*d^2+63*a*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/105*(c^2-d^2)*(15*b*c^2+56*a*c*d+25*b*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2),x,7,-2/5*b*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/f-2/15*(3*b*c+5*a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/f+2/15*(20*a*c*d+3*b*(c^2+3*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/15*(3*b*c+5*a*d)*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(1/2),x,6,-2/3*b*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/f+2/3*(b*c+3*a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/3*b*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))/(c+d*sin(e+f*x))^(1/2),x,5,2*b*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2*(b*c-a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))/(c+d*sin(e+f*x))^(3/2),x,6,-2*(b*c-a*d)*cos(e+f*x)/((c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))-2*(b*c-a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*(c^2-d^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2*b*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))/(c+d*sin(e+f*x))^(5/2),x,7,-2/3*(b*c-a*d)*cos(e+f*x)/((c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))+2/3*(4*a*c*d-b*(c^2+3*d^2))*cos(e+f*x)/((c^2-d^2)^2*f*sqrt(c+d*sin(e+f*x)))+2/3*(4*a*c*d-b*(c^2+3*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*(c^2-d^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/3*(b*c-a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))/(c+d*sin(e+f*x))^(7/2),x,8,-2/5*(b*c-a*d)*cos(e+f*x)/((c^2-d^2)*f*(c+d*sin(e+f*x))^(5/2))-2/15*(3*b*c^2-8*a*c*d+5*b*d^2)*cos(e+f*x)/((c^2-d^2)^2*f*(c+d*sin(e+f*x))^(3/2))-2/15*(3*b*c^3-23*a*c^2*d+29*b*c*d^2-9*a*d^3)*cos(e+f*x)/((c^2-d^2)^3*f*sqrt(c+d*sin(e+f*x)))-2/15*(3*b*c^3-23*a*c^2*d+29*b*c*d^2-9*a*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d*(c^2-d^2)^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/15*(3*b*c^2-8*a*c*d+5*b*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d*(c^2-d^2)^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(5/2),x,9,-2/315*(7*(9*a^2+7*b^2)*d^2-10*b*c*(b*c-9*a*d))*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f)+4/63*b*(b*c-9*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d*f)-2/9*b^2*cos(e+f*x)*(c+d*sin(e+f*x))^(7/2)/(d*f)-4/315*(84*a^2*c*d^2+15*a*b*d*(3*c^2+5*d^2)-b^2*(5*c^3-57*c*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f)+2/315*(21*a^2*d^2*(23*c^2+9*d^2)+30*a*b*d*(3*c^3+29*c*d^2)-b^2*(10*c^4-279*c^2*d^2-147*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/315*(c^2-d^2)*(5*b^2*c^3-45*a*b*c^2*d-84*a^2*c*d^2-57*b^2*c*d^2-75*a*b*d^3)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(3/2),x,8,4/35*b*(b*c-7*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f)-2/7*b^2*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d*f)-2/105*(5*(7*a^2+5*b^2)*d^2-6*b*c*(b*c-7*a*d))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f)+4/105*(70*a^2*c*d^2+21*a*b*d*(c^2+3*d^2)-b^2*(3*c^3-41*c*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/105*(c^2-d^2)*(42*a*b*c*d+35*a^2*d^2-b^2*(6*c^2-25*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(1/2),x,7,-2/5*b^2*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d*f)+4/15*b*(b*c-5*a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f)+2/15*(3*(5*a^2+3*b^2)*d^2-2*b*c*(b*c-5*a*d))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/15*b*(b*c-5*a*d)*(c^2-d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2/(c+d*sin(e+f*x))^(1/2),x,6,-2/3*b^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f)-4/3*b*(b*c-3*a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/3*((3*a^2+b^2)*d^2+2*b*c*(b*c-3*a*d))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2/(c+d*sin(e+f*x))^(3/2),x,6,2*(b*c-a*d)^2*cos(e+f*x)/(d*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))+2*(2*b^2*c^2-2*a*b*c*d+(a^2-b^2)*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*(c^2-d^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-4*b*(b*c-a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2/(c+d*sin(e+f*x))^(5/2),x,7,2/3*(b*c-a*d)^2*cos(e+f*x)/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))-4/3*(b*c-a*d)*(2*a*c*d+b*(c^2-3*d^2))*cos(e+f*x)/(d*(c^2-d^2)^2*f*sqrt(c+d*sin(e+f*x)))-4/3*(b*c-a*d)*(2*a*c*d+b*(c^2-3*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*(c^2-d^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/3*(2*a*b*c*d-a^2*d^2+b^2*(2*c^2-3*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^2/(c+d*sin(e+f*x))^(7/2),x,8,2/5*(b*c-a*d)^2*cos(e+f*x)/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^(5/2))-4/15*(b*c-a*d)*(4*a*c*d+b*(c^2-5*d^2))*cos(e+f*x)/(d*(c^2-d^2)^2*f*(c+d*sin(e+f*x))^(3/2))+2/15*(a^2*d^2*(23*c^2+9*d^2)-a*b*(6*c^3*d+58*c*d^3)-b^2*(2*c^4-19*c^2*d^2-15*d^4))*cos(e+f*x)/(d*(c^2-d^2)^3*f*sqrt(c+d*sin(e+f*x)))+2/15*(a^2*d^2*(23*c^2+9*d^2)-a*b*(6*c^3*d+58*c*d^3)-b^2*(2*c^4-19*c^2*d^2-15*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^2*(c^2-d^2)^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))+4/15*(b*c-a*d)*(4*a*c*d+b*(c^2-5*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^2*(c^2-d^2)^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^(5/2),x,10,-2/3465*(1485*a^2*b*c*d^2+693*a^3*d^3-33*a*b^2*d*(10*c^2-49*d^2)+5*b^3*(8*c^3+67*c*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f)+2/693*b*(66*a*b*c*d-297*a^2*d^2-b^2*(8*c^2+81*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d^2*f)+8/99*b^2*(b*c-6*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(7/2)/(d^2*f)-2/11*b^2*cos(e+f*x)*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(7/2)/(d*f)-2/3465*(1848*a^3*c*d^3+495*a^2*b*d^2*(3*c^2+5*d^2)-66*a*b^2*d*(5*c^3-57*c*d^2)+5*b^3*(8*c^4+57*c^2*d^2+135*d^4))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f)+2/3465*(231*a^3*d^3*(23*c^2+9*d^2)+495*a^2*b*c*d^2*(3*c^2+29*d^2)-33*a*b^2*d*(10*c^4-279*c^2*d^2-147*d^4)+5*b^3*(8*c^5+51*c^3*d^2+741*c*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/3465*(c^2-d^2)*(1848*a^3*c*d^3+495*a^2*b*d^2*(3*c^2+5*d^2)-66*a*b^2*d*(5*c^3-57*c*d^2)+5*b^3*(8*c^4+57*c^2*d^2+135*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^(3/2),x,9,2/315*b*(54*a*b*c*d-189*a^2*d^2-b^2*(8*c^2+49*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f)+8/63*b^2*(b*c-5*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(d^2*f)-2/9*b^2*cos(e+f*x)*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(5/2)/(d*f)-2/315*(189*a^2*b*c*d^2+105*a^3*d^3-9*a*b^2*d*(6*c^2-25*d^2)+b^3*(8*c^3+39*c*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f)+2/315*(420*a^3*c*d^3+189*a^2*b*d^2*(c^2+3*d^2)-a*b^2*(54*c^3*d-738*c*d^3)+b^3*(8*c^4+33*c^2*d^2+147*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/315*(c^2-d^2)*(189*a^2*b*c*d^2+105*a^3*d^3-9*a*b^2*d*(6*c^2-25*d^2)+b^3*(8*c^3+39*c*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^(1/2),x,8,8/35*b^2*(b*c-4*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(d^2*f)-2/7*b^2*cos(e+f*x)*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2)/(d*f)+2/105*b*(42*a*b*c*d-105*a^2*d^2-b^2*(8*c^2+25*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f)+2/105*(105*a^2*b*c*d^2+105*a^3*d^3-21*a*b^2*d*(2*c^2-9*d^2)+b^3*(8*c^3+19*c*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/105*b*(c^2-d^2)*(42*a*b*c*d-105*a^2*d^2-b^2*(8*c^2+25*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x))^(1/2),x,7,8/15*b^2*(b*c-3*a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f)-2/5*b^2*cos(e+f*x)*(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(d*f)-2/15*b*(30*a*b*c*d-45*a^2*d^2-b^2*(8*c^2+9*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/15*(45*a^2*b*c*d^2-15*a^3*d^3-15*a*b^2*d*(2*c^2+d^2)+b^3*(8*c^3+7*c*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x))^(3/2),x,7,2*(b*c-a*d)^2*cos(e+f*x)*(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))+2/3*b*(6*a*b*c*d-3*a^2*d^2-b^2*(4*c^2-d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*(c^2-d^2)*f)-2/3*(9*a^2*b*c*d^2-3*a^3*d^3-9*a*b^2*d*(2*c^2-d^2)+b^3*(8*c^3-5*c*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*(c^2-d^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/3*b*(18*a*b*c*d-9*a^2*d^2-b^2*(8*c^2+d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x))^(5/2),x,7,2/3*(b*c-a*d)^2*cos(e+f*x)*(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))+8/3*(b*c-a*d)^2*(a*c*d+b*(c^2-2*d^2))*cos(e+f*x)/(d^2*(c^2-d^2)^2*f*sqrt(c+d*sin(e+f*x)))+2/3*(4*a^3*c*d^3-6*a*b^2*c*d*(c^2-3*d^2)-3*a^2*b*d^2*(c^2+3*d^2)+b^3*(8*c^4-15*c^2*d^2+3*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*(c^2-d^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/3*(b*c-a*d)*(2*a*b*c*d-a^2*d^2+b^2*(8*c^2-9*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x))^(7/2),x,8,2/5*(b*c-a*d)^2*cos(e+f*x)*(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^(5/2))+8/15*(b*c-a*d)^2*(2*a*c*d+b*(c^2-3*d^2))*cos(e+f*x)/(d^2*(c^2-d^2)^2*f*(c+d*sin(e+f*x))^(3/2))-2/15*(b*c-a*d)*(a^2*d^2*(23*c^2+9*d^2)+2*a*b*d*(7*c^3-39*c*d^2)+b^2*(8*c^4-21*c^2*d^2+45*d^4))*cos(e+f*x)/(d^2*(c^2-d^2)^3*f*sqrt(c+d*sin(e+f*x)))-2/15*(b*c-a*d)*(a^2*d^2*(23*c^2+9*d^2)+2*a*b*d*(7*c^3-39*c*d^2)+b^2*(8*c^4-21*c^2*d^2+45*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*(c^2-d^2)^3*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/15*(8*a^3*c*d^3-6*a*b^2*c*d*(c^2-5*d^2)-3*a^2*b*d^2*(3*c^2+5*d^2)-b^3*(8*c^4-15*c^2*d^2+15*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*(c^2-d^2)^2*f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^3/(c+d*sin(e+f*x))^(9/2),x,9,2/7*(b*c-a*d)^2*cos(e+f*x)*(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^(7/2))+8/35*(b*c-a*d)^2*(3*a*c*d+b*(c^2-4*d^2))*cos(e+f*x)/(d^2*(c^2-d^2)^2*f*(c+d*sin(e+f*x))^(5/2))-2/105*(b*c-a*d)*(a^2*d^2*(71*c^2+25*d^2)+a*b*(26*c^3*d-218*c*d^3)+b^2*(8*c^4-17*c^2*d^2+105*d^4))*cos(e+f*x)/(d^2*(c^2-d^2)^3*f*(c+d*sin(e+f*x))^(3/2))+2/105*(16*a^3*c*d^3*(11*c^2+13*d^2)-6*a*b^2*c*d*(3*c^4-62*c^2*d^2-133*d^4)-9*a^2*b*d^2*(5*c^4+102*c^2*d^2+21*d^4)-b^3*(8*c^6-23*c^4*d^2+294*c^2*d^4+105*d^6))*cos(e+f*x)/(d^2*(c^2-d^2)^4*f*sqrt(c+d*sin(e+f*x)))+2/105*(16*a^3*c*d^3*(11*c^2+13*d^2)-6*a*b^2*c*d*(3*c^4-62*c^2*d^2-133*d^4)-9*a^2*b*d^2*(5*c^4+102*c^2*d^2+21*d^4)-b^3*(8*c^6-23*c^4*d^2+294*c^2*d^4+105*d^6))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(d^3*(c^2-d^2)^4*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/105*(b*c-a*d)*(a^2*d^2*(71*c^2+25*d^2)+a*b*(26*c^3*d-218*c*d^3)+b^2*(8*c^4-17*c^2*d^2+105*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(d^3*(c^2-d^2)^3*f*sqrt(c+d*sin(e+f*x)))],

# m<0
[(c+d*sin(e+f*x))^(5/2)/(a+b*sin(e+f*x)),x,9,-2/3*d^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*f)+2/3*d*(7*b*c-3*a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))-2/3*d*(6*a*b*c*d-3*a^2*d^2-b^2*(2*c^2+d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^3*f*sqrt(c+d*sin(e+f*x)))+2*(b*c-a*d)^3*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^3*(a+b)*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+b*sin(e+f*x)),x,8,2*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2*d*(b*c-a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^2*f*sqrt(c+d*sin(e+f*x)))+2*(b*c-a*d)^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^2*(a+b)*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x)),x,5,2*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b*f*sqrt(c+d*sin(e+f*x)))+2*(b*c-a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b*(a+b)*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(1/2)),x,2,2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a+b)*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2)),x,7,-2*d^2*cos(e+f*x)/((b*c-a*d)*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))-2*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((b*c-a*d)*(c^2-d^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2*b*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a+b)*(b*c-a*d)*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(5/2)),x,10,-2/3*d^2*cos(e+f*x)/((b*c-a*d)*(c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))-2/3*d^2*(7*b*c^2-4*a*c*d-3*b*d^2)*cos(e+f*x)/((b*c-a*d)^2*(c^2-d^2)^2*f*sqrt(c+d*sin(e+f*x)))-2/3*d*(7*b*c^2-4*a*c*d-3*b*d^2)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((b*c-a*d)^2*(c^2-d^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+2/3*d*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((b*c-a*d)*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))+2*b^2*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a+b)*(b*c-a*d)^2*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(7/2)/(a+b*sin(e+f*x))^2,x,10,(b*c-a*d)^2*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(b*(a^2-b^2)*f*(a+b*sin(e+f*x)))+1/3*d*(6*a*b*c*d-5*a^2*d^2-b^2*(3*c^2-2*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b^2*(a^2-b^2)*f)+1/3*(29*a^2*b*c*d^2-15*a^3*d^3+b^3*(3*c^3-20*c*d^2)-a*b^2*(9*c^2*d-12*d^3))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b^3*(a^2-b^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-1/3*(24*a^3*b*c*d^3-15*a^4*d^4-12*a*b^3*c*d*(c^2+3*d^2)+2*a^2*b^2*d^2*(c^2+8*d^2)+b^4*(3*c^4+16*c^2*d^2+2*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^4*(a^2-b^2)*f*sqrt(c+d*sin(e+f*x)))+(b*c-a*d)^3*(2*a*b*c+5*a^2*d-7*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)*b^4*(a+b)^2*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(5/2)/(a+b*sin(e+f*x))^2,x,9,(b*c-a*d)^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*(a+b*sin(e+f*x)))-(2*a*b*c*d-3*a^2*d^2-b^2*(c^2-2*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b^2*(a^2-b^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+(b*c-a*d)*(2*a*b*c*d+3*a^2*d^2-b^2*(c^2+4*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^3*(a^2-b^2)*f*sqrt(c+d*sin(e+f*x)))+(b*c-a*d)^2*(2*a*b*c+3*a^2*d-5*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)*b^3*(a+b)^2*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+b*sin(e+f*x))^2,x,9,(b*c-a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*f*(a+b*sin(e+f*x)))+(b*c-a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))+(2*a*b*c*d+a^2*d^2-b^2*(c^2+2*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^2*(a^2-b^2)*f*sqrt(c+d*sin(e+f*x)))+(b*c-a*d)*(2*a*b*c+a^2*d-3*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)*b^2*(a+b)^2*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x))^2,x,9,b*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*f*(a+b*sin(e+f*x)))+sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-(b*c-a*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b*(a^2-b^2)*f*sqrt(c+d*sin(e+f*x)))+(2*a*b*c-a^2*d-b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)*b*(a+b)^2*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(1/2)),x,9,b^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x)))+b*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*(b*c-a*d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a^2-b^2)*f*sqrt(c+d*sin(e+f*x)))+(2*a*b*c-3*a^2*d+b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)*(a+b)^2*(b*c-a*d)*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(3/2)),x,10,d*(2*a^2*d^2+b^2*(c^2-3*d^2))*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)^2*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))+b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))+(2*a^2*d^2+b^2*(c^2-3*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*(b*c-a*d)^2*(c^2-d^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-b*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a^2-b^2)*(b*c-a*d)*f*sqrt(c+d*sin(e+f*x)))+b*(2*a*b*c-5*a^2*d+3*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)*(a+b)^2*(b*c-a*d)^2*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(5/2)),x,11,1/3*d*(2*a^2*d^2+b^2*(3*c^2-5*d^2))*cos(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))+b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(3/2))-1/3*(8*a^3*c*d^4-8*a*b^2*c*d^4-4*a^2*b*d^3*(5*c^2-3*d^2)-b^3*(3*c^4*d-26*c^2*d^3+15*d^5))*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)^3*(c^2-d^2)^2*f*sqrt(c+d*sin(e+f*x)))-1/3*(8*a^3*c*d^3-8*a*b^2*c*d^3-4*a^2*b*d^2*(5*c^2-3*d^2)-b^3*(3*c^4-26*c^2*d^2+15*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*(b*c-a*d)^3*(c^2-d^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))-1/3*(2*a^2*d^2+b^2*(3*c^2-5*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a^2-b^2)*(b*c-a*d)^2*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))+b^2*(2*a*b*c-7*a^2*d+5*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)*(a+b)^2*(b*c-a*d)^3*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(9/2)/(a+b*sin(e+f*x))^3,x,11,1/4*(b*c-a*d)^2*(6*a*b*c+7*a^2*d-13*b^2*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(b^2*(a^2-b^2)^2*f*(a+b*sin(e+f*x)))+1/2*(b*c-a*d)^2*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)/(b*(a^2-b^2)*f*(a+b*sin(e+f*x))^2)+1/12*d*(36*a^3*b*c*d^2-35*a^4*d^3+b^4*d*(45*c^2-8*d^2)-18*a*b^3*c*(c^2+5*d^2)+a^2*b^2*d*(9*c^2+61*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b^3*(a^2-b^2)^2*f)+1/12*(185*a^4*b*c*d^3-105*a^5*d^4-b^5*c*d*(51*c^2-104*d^2)-15*a^3*b^2*d^2*(3*c^2-13*d^2)-a^2*b^3*c*d*(21*c^2+361*d^2)+9*a*b^4*(2*c^4+17*c^2*d^2-8*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b^4*(a^2-b^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))-1/12*(150*a^5*b*c*d^4-105*a^6*d^5-12*a^3*b^3*c*d^2*(4*c^2+29*d^2)+a^4*b^2*d^3*(26*c^2+223*d^2)-b^6*d*(57*c^4+136*c^2*d^2+8*d^4)+6*a*b^5*c*(3*c^4+38*c^2*d^2+48*d^4)-a^2*b^4*d*(33*c^4+70*c^2*d^2+128*d^4))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^5*(a^2-b^2)^2*f*sqrt(c+d*sin(e+f*x)))+1/4*(b*c-a*d)^3*(20*a^3*b*c*d-44*a*b^3*c*d+35*a^4*d^2+2*a^2*b^2*(4*c^2-43*d^2)+b^4*(4*c^2+63*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)^2*b^5*(a+b)^3*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(7/2)/(a+b*sin(e+f*x))^3,x,10,1/2*(b*c-a*d)^2*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(b*(a^2-b^2)*f*(a+b*sin(e+f*x))^2)+1/4*(b*c-a*d)^2*(6*a*b*c+5*a^2*d-11*b^2*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b^2*(a^2-b^2)^2*f*(a+b*sin(e+f*x)))-1/4*(8*a^3*b*c*d^2-15*a^4*d^3+b^4*d*(13*c^2-8*d^2)-2*a*b^3*c*(3*c^2+13*d^2)+a^2*b^2*d*(5*c^2+29*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b^3*(a^2-b^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+3/4*(b*c-a*d)*(4*a^3*b*c*d^2+5*a^4*d^3+a^2*b^2*d*(c^2-11*d^2)-2*a*b^3*c*(c^2+5*d^2)+b^4*d*(5*c^2+8*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^4*(a^2-b^2)^2*f*sqrt(c+d*sin(e+f*x)))+1/4*(b*c-a*d)^2*(12*a^3*b*c*d-36*a*b^3*c*d+15*a^4*d^2+2*a^2*b^2*(4*c^2-19*d^2)+b^4*(4*c^2+35*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)^2*b^4*(a+b)^3*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(5/2)/(a+b*sin(e+f*x))^3,x,10,1/2*(b*c-a*d)^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*(a+b*sin(e+f*x))^2)+3/4*(b*c-a*d)*(2*a*b*c+a^2*d-3*b^2*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)^2*f*(a+b*sin(e+f*x)))+3/4*(b*c-a*d)*(2*a*b*c+a^2*d-3*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b^2*(a^2-b^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))+1/4*(4*a^3*b*c*d^2+3*a^4*d^3+a^2*b^2*d*(7*c^2-5*d^2)+b^4*d*(11*c^2+8*d^2)-2*a*b^3*c*(3*c^2+11*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^3*(a^2-b^2)^2*f*sqrt(c+d*sin(e+f*x)))+1/4*(b*c-a*d)*(4*a^3*b*c*d-28*a*b^3*c*d+3*a^4*d^2+2*a^2*b^2*(4*c^2-3*d^2)+b^4*(4*c^2+15*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)^2*b^3*(a+b)^3*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+b*sin(e+f*x))^3,x,10,1/2*(b*c-a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*f*(a+b*sin(e+f*x))^2)+1/4*(6*a*b*c-a^2*d-5*b^2*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)^2*f*(a+b*sin(e+f*x)))+1/4*(6*a*b*c-a^2*d-5*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))-1/4*(b*c-a*d)*(6*a*b*c+a^2*d-7*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b^2*(a^2-b^2)^2*f*sqrt(c+d*sin(e+f*x)))-1/4*(4*a^3*b*c*d+20*a*b^3*c*d+a^4*d^2-b^4*(4*c^2+3*d^2)-2*a^2*b^2*(4*c^2+5*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)^2*b^2*(a+b)^3*f*sqrt(c+d*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x))^3,x,10,1/2*b*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*f*(a+b*sin(e+f*x))^2)+1/4*b*(6*a*b*c-5*a^2*d-b^2*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)^2*(b*c-a*d)*f*(a+b*sin(e+f*x)))+1/4*(6*a*b*c-5*a^2*d-b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((a^2-b^2)^2*(b*c-a*d)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-3/4*(2*a*b*c-a^2*d-b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/(b*(a^2-b^2)^2*f*sqrt(c+d*sin(e+f*x)))-1/4*(12*a^3*b*c*d+12*a*b^3*c*d-3*a^4*d^2-b^4*(4*c^2-d^2)-2*a^2*b^2*(4*c^2+5*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)^2*b*(a+b)^3*(b*c-a*d)*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^(1/2)),x,10,1/2*b^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))^2)+3/4*b^2*(2*a*b*c-3*a^2*d+b^2*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)^2*(b*c-a*d)^2*f*(a+b*sin(e+f*x)))+3/4*b*(2*a*b*c-3*a^2*d+b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((a^2-b^2)^2*(b*c-a*d)^2*f*sqrt((c+d*sin(e+f*x))/(c+d)))-1/4*(6*a*b*c-7*a^2*d+b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a^2-b^2)^2*(b*c-a*d)*f*sqrt(c+d*sin(e+f*x)))-1/4*(20*a^3*b*c*d+4*a*b^3*c*d-15*a^4*d^2-2*a^2*b^2*(4*c^2-3*d^2)-b^4*(4*c^2+3*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)^2*(a+b)^3*(b*c-a*d)^2*f*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^3*(c+d*sin(e+f*x))^(3/2)),x,11,-1/4*d*(8*a^4*d^3+a^2*b^2*d*(13*c^2-29*d^2)-b^4*d*(7*c^2-15*d^2)-6*a*b^3*c*(c^2-d^2))*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^3*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))+1/2*b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))^2*sqrt(c+d*sin(e+f*x)))+1/4*b^2*(6*a*b*c-11*a^2*d+5*b^2*d)*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^2*f*(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))-1/4*(8*a^4*d^3+a^2*b^2*d*(13*c^2-29*d^2)-b^4*d*(7*c^2-15*d^2)-6*a*b^3*c*(c^2-d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticE(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt(c+d*sin(e+f*x))/((a^2-b^2)^2*(b*c-a*d)^3*(c^2-d^2)*f*sqrt((c+d*sin(e+f*x))/(c+d)))-1/4*b*(6*a*b*c-11*a^2*d+5*b^2*d)*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticF(sin(1/2*(e-1/2*Pi+f*x)),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a^2-b^2)^2*(b*c-a*d)^2*f*sqrt(c+d*sin(e+f*x)))-1/4*b*(28*a^3*b*c*d-4*a*b^3*c*d-35*a^4*d^2-2*a^2*b^2*(4*c^2-19*d^2)-b^4*(4*c^2+15*d^2))*sqrt(cos(1/2*(e-1/2*Pi+f*x))^2)/cos(1/2*(e-1/2*Pi+f*x))*EllipticPi(sin(1/2*(e-1/2*Pi+f*x)),2*b/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sin(e+f*x))/(c+d))/((a-b)^2*(a+b)^3*(b*c-a*d)^3*f*sqrt(c+d*sin(e+f*x)))],

#  {1/(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2), x, 12, -((d*(8*a^4*d^3 + a^2*b^2*d*(45*c^2 - 61*d^2) - b^4*d*(27*c^2 - 35*d^2) - 18*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(12*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2))) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)) + (b^2*(6*a*b*c - 13*a^2*d + 7*b^2*d)*Cos[e + f*x])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) + (d*(32*a^5*c*d^4 - 64*a^3*b^2*c*d^4 - 8*a^4*b*d^3*(13*c^2 - 9*d^2) + 2*a*b^4*c*(9*c^4 - 18*c^2*d^2 + 25*d^4) + b^5*d*(33*c^4 - 170*c^2*d^2 + 105*d^4) - a^2*b^3*d*(51*c^4 - 310*c^2*d^2 + 195*d^4))*Cos[e + f*x])/(12*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((32*a^5*c*d^4 - 64*a^3*b^2*c*d^4 - 8*a^4*b*d^3*(13*c^2 - 9*d^2) + 2*a*b^4*c*(9*c^4 - 18*c^2*d^2 + 25*d^4) + b^5*d*(33*c^4 - 170*c^2*d^2 + 105*d^4) - a^2*b^3*d*(51*c^4 - 310*c^2*d^2 + 195*d^4))*EllipticE[Pi/4 + (1/2)*(-e - f*x), (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(12*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((8*a^4*d^3 + a^2*b^2*d*(45*c^2 - 61*d^2) - b^4*d*(27*c^2 - 35*d^2) - 18*a*b^3*c*(c^2 - d^2))*EllipticF[Pi/4 + (1/2)*(-e - f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(12*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (b^2*(36*a^3*b*c*d - 12*a*b^3*c*d - 63*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 43*d^2) - b^4*(4*c^2 + 35*d^2))*EllipticPi[(2*b)/(a + b), -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^4*f*Sqrt[c + d*Sin[e + f*x]])} 

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

# m>0
[(a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(5/2),x,8,-1/8*(5*a^2*b*c*d^2-a^3*d^3-a*b^2*d*(15*c^2+4*d^2)-5*b^3*(c^3+4*c*d^2))*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*d*f*sqrt(a+b))+1/24*(c-d)*(14*a*b*c*d-3*a^2*d^2+b^2*(33*c^2+16*d^2))*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(a+b)*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^2*(b*c-a*d)*f)+1/24*(a+b)^(3/2)*(3*a^2*d^2-6*a*b*d*(2*c+d)+b^2*(33*c^2+26*c*d+16*d^2))*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))))/(b^3*f*sqrt(c+d))-1/3*d^2*cos(e+f*x)*(a+b*sin(e+f*x))^(3/2)*sqrt(c+d*sin(e+f*x))/(b*f)-1/24*(14*a*b*c*d-3*a^2*d^2+b^2*(33*c^2+16*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*f*sqrt(a+b*sin(e+f*x)))-1/12*d*(13*b*c-3*a*d)*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(b*f)],
[(a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(3/2),x,8,1/4*(6*a*b*c*d-a^2*d^2+b^2*(3*c^2+4*d^2))*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^2*d*f*sqrt(a+b))+1/4*(c-d)*(5*b*c+a*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(a+b)*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*(b*c-a*d)*f)-1/2*b*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)/(f*sqrt(a+b*sin(e+f*x)))+1/4*(a+b)^(3/2)*(5*b*c-a*d+2*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))))/(b^2*f*sqrt(c+d))+1/2*(b*c-a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+b*sin(e+f*x)))-1/4*(5*b*c+a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+b*sin(e+f*x)))],
[(a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(1/2),x,7,(b*c+a*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*d*f*sqrt(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(a+b)*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*c-a*d)*f)+(a+b)^(3/2)*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))))/(b*f*sqrt(c+d))-b*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+b*sin(e+f*x)))],
[(a+b*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^(1/2),x,1,2*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))))/(d*f*sqrt(a+b))],
[(a+b*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^(3/2),x,3,-2*(a-b)*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(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))))/((c-d)*(b*c-a*d)*f*sqrt(c+d))+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))))/((c-d)*(b*c-a*d)*f*sqrt(c+d))],
[(a+b*sin(e+f*x))^(1/2)/(c+d*sin(e+f*x))^(5/2),x,4,2/3*d*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/((c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))+2/3*(a-b)*(4*a*c*d-b*(3*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(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))))/((c-d)^2*(c+d)^(3/2)*(b*c-a*d)^2*f)+2/3*(a-b)*(3*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))))/((c-d)^2*(c+d)^(3/2)*(b*c-a*d)*f)],
[(a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(5/2),x,9,-1/64*(20*a^3*b*c*d^3-3*a^4*d^4-60*a*b^3*c*d*(c^2+4*d^2)-6*a^2*b^2*d^2*(15*c^2+4*d^2)+b^4*(5*c^4-120*c^2*d^2-48*d^4))*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*d^2*f*sqrt(a+b))+1/192*(c-d)*(57*a^2*b*c*d^2-9*a^3*d^3+a*b^2*d*(337*c^2+156*d^2)+b^3*(15*c^3+284*c*d^2))*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(a+b)*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^2*d*(b*c-a*d)*f)+1/192*(a+b)^(3/2)*(9*a^3*d^3-3*a^2*b*d^2*(17*c+6*d)+3*a*b^2*d*(73*c^2+36*c*d+28*d^2)+b^3*(15*c^3+118*c^2*d+284*c*d^2+72*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))))/(b^3*d*f*sqrt(c+d))-1/24*d*(17*b*c-3*a*d)*cos(e+f*x)*(a+b*sin(e+f*x))^(3/2)*sqrt(c+d*sin(e+f*x))/(b*f)-1/4*d^2*cos(e+f*x)*(a+b*sin(e+f*x))^(5/2)*sqrt(c+d*sin(e+f*x))/(b*f)-1/192*(57*a^2*b*c*d^2-9*a^3*d^3+a*b^2*d*(337*c^2+156*d^2)+b^3*(15*c^3+284*c*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*d*f*sqrt(a+b*sin(e+f*x)))-1/96*(54*a*b*c*d-9*a^2*d^2+b^2*(59*c^2+36*d^2))*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(b*f)],
[(a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(3/2),x,8,1/8*(b*c+a*d)*(10*a*b*c*d-a^2*d^2-b^2*(c^2-12*d^2))*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^2*d^2*f*sqrt(a+b))+1/24*(c-d)*(38*a*b*c*d+3*a^2*d^2+b^2*(3*c^2+16*d^2))*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(a+b)*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*d*(b*c-a*d)*f)-1/3*b*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)*sqrt(a+b*sin(e+f*x))/f-1/24*(a+b)^(3/2)*(3*a^2*d^2-6*a*b*d*(4*c+d)-b^2*(3*c^2+14*c*d+16*d^2))*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))))/(b^2*d*f*sqrt(c+d))-1/24*(38*a*b*c*d+3*a^2*d^2+b^2*(3*c^2+16*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f*sqrt(a+b*sin(e+f*x)))-1/12*(3*b*c+7*a*d)*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/f],
[(a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(1/2),x,7,1/4*(6*a*b*c*d+3*a^2*d^2-b^2*(c^2-4*d^2))*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*d^2*f*sqrt(a+b))+1/4*(c-d)*(b*c+5*a*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(a+b)*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))))/(d*(b*c-a*d)*f)+1/4*(a+b)^(3/2)*(3*a*d+b*(c+2*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))))/(b*d*f*sqrt(c+d))-1/4*b*(b*c+5*a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d*f*sqrt(a+b*sin(e+f*x)))-1/2*b*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/f],
[(a+b*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^(1/2),x,6,(b*(c-d)-2*a*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))))/(d^2*f*sqrt(c+d))-(b*c-3*a*d)*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))))/(d^2*f*sqrt(c+d))-(a-b)*b*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(a+b)*sqrt(c+d)*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))))/(d*(b*c-a*d)*f)-b*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/(f*sqrt(c+d*sin(e+f*x)))],
[(a+b*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^(3/2),x,5,2*(a-b)*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(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))))/((c-d)*d*f*sqrt(c+d))-2*(b*(c-2*d)+a*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))))/((c-d)*d^2*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))))/(d^2*f*sqrt(c+d))],
[(a+b*sin(e+f*x))^(3/2)/(c+d*sin(e+f*x))^(5/2),x,4,-2/3*(b*c-a*d)*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/((c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))-8/3*(a-b)*(a*c-b*d)*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(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))))/((c-d)^2*(c+d)^(3/2)*(b*c-a*d)*f)+2/3*(a-b)*(a*(3*c+d)-b*(c+3*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))))/((c-d)^2*(c+d)^(3/2)*(b*c-a*d)*f)],
[(a+b*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(5/2),x,10,-1/128*(b*c+a*d)*(28*a^3*b*c*d^3-3*a^4*d^4+28*a*b^3*c*d*(c^2-20*d^2)-2*a^2*b^2*d^2*(89*c^2+20*d^2)-b^4*(3*c^4+40*c^2*d^2+240*d^4))*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*d^3*f*sqrt(a+b))+1/1920*(c-d)*(360*a^3*b*c*d^3-45*a^4*d^4+2*a^2*b^2*d^2*(1877*c^2+846*d^2)+8*a*b^3*d*(45*c^3+791*c*d^2)-b^4*(45*c^4-1692*c^2*d^2-1024*d^4))*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(a+b)*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^2*d^2*(b*c-a*d)*f)-1/240*(110*a*b*c*d+93*a^2*d^2-b^2*(15*c^2-64*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)*sqrt(a+b*sin(e+f*x))/(d*f)+3/40*b*(b*c-7*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)*sqrt(a+b*sin(e+f*x))/(d*f)-1/5*b^2*cos(e+f*x)*(c+d*sin(e+f*x))^(7/2)*sqrt(a+b*sin(e+f*x))/(d*f)+1/1920*(a+b)^(3/2)*(45*a^4*d^4-30*a^3*b*d^3*(11*c+3*d)+30*a^2*b^2*d^2*(64*c^2+23*c*d+22*d^2)+2*a*b^3*d*(165*c^3+917*c^2*d+2392*c*d^2+516*d^3)-b^4*(45*c^4-30*c^3*d-1692*c^2*d^2-1544*c*d^3-1024*d^4))*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))))/(b^3*d^2*f*sqrt(c+d))-1/1920*(360*a^3*b*c*d^3-45*a^4*d^4+2*a^2*b^2*d^2*(1877*c^2+846*d^2)+8*a*b^3*d*(45*c^3+791*c*d^2)-b^4*(45*c^4-1692*c^2*d^2-1024*d^4))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*d^2*f*sqrt(a+b*sin(e+f*x)))-1/960*(917*a^2*b*c*d^2+15*a^3*d^3+a*b^2*d*(345*c^2+772*d^2)-b^3*(45*c^3-516*c*d^2))*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(b*d*f)],
[(a+b*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(3/2),x,9,1/64*(60*a^3*b*c*d^3-5*a^4*d^4-20*a*b^3*c*d*(c^2-12*d^2)+3*b^4*(c^2+4*d^2)^2+30*a^2*b^2*d^2*(3*c^2+4*d^2))*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^2*d^3*f*sqrt(a+b))+1/192*(c-d)*(337*a^2*b*c*d^2+15*a^3*d^3+a*b^2*d*(57*c^2+284*d^2)-b^3*(9*c^3-156*c*d^2))*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(a+b)*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*d^2*(b*c-a*d)*f)+1/24*b*(3*b*c-17*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)*sqrt(a+b*sin(e+f*x))/(d*f)-1/4*b^2*cos(e+f*x)*(c+d*sin(e+f*x))^(5/2)*sqrt(a+b*sin(e+f*x))/(d*f)-1/192*(a+b)^(3/2)*(15*a^3*d^3-15*a^2*b*d^2*(11*c+2*d)-a*b^2*d*(51*c^2+172*c*d+212*d^2)+b^3*(9*c^3-6*c^2*d-156*c*d^2-72*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))))/(b^2*d^2*f*sqrt(c+d))-1/192*(337*a^2*b*c*d^2+15*a^3*d^3+a*b^2*d*(57*c^2+284*d^2)-b^3*(9*c^3-156*c*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt(a+b*sin(e+f*x)))-1/96*(54*a*b*c*d+59*a^2*d^2-9*b^2*(c^2-4*d^2))*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(d*f)],
[(a+b*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(1/2),x,8,1/8*(15*a^2*b*c*d^2+5*a^3*d^3-5*a*b^2*d*(c^2-4*d^2)+b^3*(c^3+4*c*d^2))*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*d^3*f*sqrt(a+b))+1/24*(c-d)*(14*a*b*c*d+33*a^2*d^2-b^2*(3*c^2-16*d^2))*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(a+b)*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))))/(d^2*(b*c-a*d)*f)-1/3*b^2*cos(e+f*x)*(c+d*sin(e+f*x))^(3/2)*sqrt(a+b*sin(e+f*x))/(d*f)+1/24*(a+b)^(3/2)*(15*a^2*d^2+6*a*b*d*(2*c+3*d)-b^2*(3*c^2-2*c*d-16*d^2))*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))))/(b*d^2*f*sqrt(c+d))-1/24*b*(14*a*b*c*d+33*a^2*d^2-b^2*(3*c^2-16*d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt(a+b*sin(e+f*x)))+1/12*b*(3*b*c-13*a*d)*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(d*f)],
[(a+b*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(1/2),x,7,-1/4*(10*a*b*c*d-15*a^2*d^2-b^2*(3*c^2+4*d^2))*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))))/(d^3*f*sqrt(a+b))-3/4*b*(c-d)*(b*c-3*a*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(a+b)*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))))/(d^2*(b*c-a*d)*f)-1/4*(a+b)^(3/2)*(3*b*c-7*a*d-2*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))))/(d^2*f*sqrt(c+d))+3/4*b^2*(b*c-3*a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*f*sqrt(a+b*sin(e+f*x)))-1/2*b^2*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(d*f)],
[(a+b*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(3/2),x,7,-(4*a*b*c*d-2*a^2*d^2-b^2*(3*c^2-d^2))*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(a+b)*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))))/(d^2*(b*c-a*d)*f*sqrt(c+d))-b*(3*b*c-5*a*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))))/(d^3*f*sqrt(a+b))-(a+b)^(3/2)*(2*a*d-b*(3*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((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))))/(d^2*(c+d)^(3/2)*f)+2*(b*c-a*d)^2*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*sqrt(c+d*sin(e+f*x)))+b*(4*a*b*c*d-2*a^2*d^2-b^2*(3*c^2-d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(d^2*(c^2-d^2)*f*sqrt(a+b*sin(e+f*x)))],
[(a+b*sin(e+f*x))^(5/2)/(c+d*sin(e+f*x))^(5/2),x,6,2/3*(b*c-a*d)^2*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/(d*(c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))+2/3*(a-b)*(3*b*c^2+4*a*c*d-7*b*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(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))))/((c-d)^2*d^2*(c+d)^(3/2)*f)-2/3*(a^2*d^2*(3*c+d)+a*b*d*(3*c^2-4*c*d-7*d^2)+b^2*(3*c^3-6*c^2*d-2*c*d^2+9*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(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))))/((c-d)^2*d^3*(c+d)^(3/2)*f)+2*b^2*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))))/(d^3*f*sqrt(c+d))],

# m<0
[(c+d*sin(e+f*x))^(5/2)/(a+b*sin(e+f*x))^(1/2),x,7,-1/4*(10*a*b*c*d-3*a^2*d^2-b^2*(15*c^2+4*d^2))*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))+3/4*(c-d)*d*(3*b*c-a*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(a+b)*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^2*(b*c-a*d)*f)+1/4*(3*a^2*d^2-a*b*d*(7*c+3*d)+b^2*(8*c^2+9*c*d+2*d^2))*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))))/(b^3*f*sqrt(c+d))-3/4*d*(3*b*c-a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*f*sqrt(a+b*sin(e+f*x)))-1/2*d^2*cos(e+f*x)*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x))/(b*f)],
[(c+d*sin(e+f*x))^(3/2)/(a+b*sin(e+f*x))^(1/2),x,6,(3*b*c-a*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^2*f*sqrt(a+b))+(c-d)*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(a+b)*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*(b*c-a*d)*f)-(a*d-b*(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))))/(b^2*f*sqrt(c+d))-d*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(f*sqrt(a+b*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x))^(1/2),x,1,2*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*f*sqrt(c+d))],
[1/((a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(1/2)),x,1,2*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))))/((b*c-a*d)*f*sqrt(c+d))],
[1/((a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(3/2)),x,3,2*(a-b)*d*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(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))))/((c-d)*(b*c-a*d)^2*f*sqrt(c+d))+2*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))))/((c-d)*(b*c-a*d)*f*sqrt(c+d))],
[1/((a+b*sin(e+f*x))^(1/2)*(c+d*sin(e+f*x))^(5/2)),x,4,-2/3*d^2*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/((b*c-a*d)*(c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))-4/3*(a-b)*d*(2*a*c*d-b*(3*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(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))))/((c-d)^2*(c+d)^(3/2)*(b*c-a*d)^3*f)-2/3*(a*d*(3*c+d)-b*(3*c^2+3*c*d-2*d^2))*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))))/((c-d)^2*(c+d)^(3/2)*(b*c-a*d)^2*f)],
[(c+d*sin(e+f*x))^(5/2)/(a+b*sin(e+f*x))^(3/2),x,7,(c-d)*(2*b^2*c^2-4*a*b*c*d+3*a^2*d^2-b^2*d^2)*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))+d*(5*b*c-3*a*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))-(3*a^2*d^2-2*a*b*d*(c+3*d)-b^2*(2*c^2-6*c*d-d^2))*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*(b*c-a*d)^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*sqrt(a+b*sin(e+f*x)))+(4*a*b*c*d-3*a^2*d^2-b^2*(2*c^2-d^2))*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*sqrt(a+b*sin(e+f*x)))],
[(c+d*sin(e+f*x))^(3/2)/(a+b*sin(e+f*x))^(3/2),x,5,2*(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*f*sqrt(a+b))+2*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^2*f*sqrt(a+b))+2*(b*(c-2*d)+a*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^2*f*sqrt(c+d))],
[(c+d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x))^(3/2),x,3,2*(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)*f*sqrt(a+b))+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*c-a*d)*f*sqrt(c+d))],
[1/((a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(1/2)),x,3,2*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*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))],
[1/((a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(3/2)),x,4,-2*(a^2*d^2+b^2*(c^2-2*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*(b*(c-2*d)-a*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^2*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)))],
[1/((a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(5/2)),x,5,2*b^2*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^2*d^2+b^2*(3*c^2-4*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*(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*(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))],
[(c+d*sin(e+f*x))^(5/2)/(a+b*sin(e+f*x))^(5/2),x,6,2/3*(c-d)*(4*a*b*c+3*a^2*d-7*b^2*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)^2*b^2*(a+b)^(3/2)*f)+2*d^2*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))+2/3*(3*a^2*b*(c-2*d)*d+3*a^3*d^2+a*b^2*(3*c^2-4*c*d-2*d^2)+b^3*(c^2-7*c*d+9*d^2))*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))))/((a-b)^2*b^3*f*sqrt(a+b)*sqrt(c+d))+2/3*(b*c-a*d)^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/(b*(a^2-b^2)*f*(a+b*sin(e+f*x))^(3/2))],
[(c+d*sin(e+f*x))^(3/2)/(a+b*sin(e+f*x))^(5/2),x,4,8/3*(c-d)*(a*c-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)^2*(a+b)^(3/2)*(b*c-a*d)*f)+2/3*(c-d)*(3*a*c+b*c-a*d-3*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))))/((a-b)^2*(b*c-a*d)*f*sqrt(a+b)*sqrt(c+d))+2/3*(b*c-a*d)*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*f*(a+b*sin(e+f*x))^(3/2))],
[(c+d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x))^(5/2),x,4,2/3*(c-d)*(4*a*b*c-3*a^2*d-b^2*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)^2*(a+b)^(3/2)*(b*c-a*d)^2*f)+2/3*(3*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((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)^2*(b*c-a*d)*f*sqrt(a+b)*sqrt(c+d))+2/3*b*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*f*(a+b*sin(e+f*x))^(3/2))],
[1/((a+b*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(1/2)),x,4,4/3*b*(c-d)*(2*a*b*c-3*a^2*d+b^2*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)^2*(a+b)^(3/2)*(b*c-a*d)^3*f)+2/3*(3*a*b*(c-d)-3*a^2*d+b^2*(c+2*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))))/((a-b)^2*(b*c-a*d)^2*f*sqrt(a+b)*sqrt(c+d))+2/3*b^2*cos(e+f*x)*sqrt(c+d*sin(e+f*x))/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))^(3/2))],
[1/((a+b*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(3/2)),x,5,2/3*(3*a^4*d^3-b^4*d*(5*c^2-8*d^2)+3*a^2*b^2*d*(3*c^2-5*d^2)-4*a*b^3*c*(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))))/((a^2-b^2)*(c-d)*(b*c-a*d)^4*f*sqrt(a+b)*sqrt(c+d))-2/3*(3*a^2*b*(2*c-3*d)*d-3*a^3*d^2-3*a*b^2*(c^2-2*d^2)+b^3*(c^2-6*c*d+8*d^2))*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))))/((a^2-b^2)*(c-d)*(b*c-a*d)^3*f*sqrt(a+b)*sqrt(c+d))+2/3*b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))^(3/2)*sqrt(c+d*sin(e+f*x)))+8/3*b^2*(a*b*c-2*a^2*d+b^2*d)*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^2*f*sqrt(a+b*sin(e+f*x))*sqrt(c+d*sin(e+f*x)))],
[1/((a+b*sin(e+f*x))^(5/2)*(c+d*sin(e+f*x))^(5/2)),x,6,2/3*b^2*cos(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(a+b*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))^(3/2))+4/3*b^2*(2*a*b*c-5*a^2*d+3*b^2*d)*cos(e+f*x)/((a^2-b^2)^2*(b*c-a*d)^2*f*(c+d*sin(e+f*x))^(3/2)*sqrt(a+b*sin(e+f*x)))-2/3*d*(a^4*d^3+a^2*b^2*d*(11*c^2-13*d^2)-b^4*d*(7*c^2-8*d^2)-4*a*b^3*c*(c^2-d^2))*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/((a^2-b^2)^2*(b*c-a*d)^3*(c^2-d^2)*f*(c+d*sin(e+f*x))^(3/2))-8/3*(a^5*c*d^4-2*a^3*b^2*c*d^4+a*b^4*c*(c^4-2*c^2*d^2+2*d^4)+b^5*d*(2*c^4-7*c^2*d^2+4*d^4)-a^2*b^3*d*(3*c^4-12*c^2*d^2+7*d^4)-a^4*b*(3*c^2*d^3-2*d^5))*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))))/((a^2-b^2)*(c-d)^2*(c+d)^(3/2)*(b*c-a*d)^5*f*sqrt(a+b))-2/3*(a^4*d^3*(3*c+d)-9*a^3*b*d^2*(c^2-d^2)+a^2*b^2*d*(9*c^3-18*c^2*d-15*c*d^2+16*d^3)+b^4*(c^4-9*c^3*d+16*c^2*d^2+12*c*d^3-16*d^4)-3*a*b^3*(c^4-5*c^2*d^2+4*d^4))*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))))/((a^2-b^2)*(c-d)^2*(c+d)^(3/2)*(b*c-a*d)^4*f*sqrt(a+b))],

# Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic
[(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x,0,Unintegrable((a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x)],
[(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^2,x,8,-d^2*cos(e+f*x)*(a+b*sin(e+f*x))^(1+m)/(b*f*(2+m))+(a+b)*d*(a*d-2*b*c*(2+m))*AppellF1(1/2,1/2,-1-m,3/2,1/2*(1-sin(e+f*x)),b*(1-sin(e+f*x))/(a+b))*cos(e+f*x)*(a+b*sin(e+f*x))^m*sqrt(2)/(b^2*f*(2+m)*((a+b*sin(e+f*x))/(a+b))^m*sqrt(1+sin(e+f*x)))-(a*d*(a*d-2*b*c*(2+m))+b^2*(d^2*(1+m)+c^2*(2+m)))*AppellF1(1/2,1/2,-m,3/2,1/2*(1-sin(e+f*x)),b*(1-sin(e+f*x))/(a+b))*cos(e+f*x)*(a+b*sin(e+f*x))^m*sqrt(2)/(b^2*f*(2+m)*((a+b*sin(e+f*x))/(a+b))^m*sqrt(1+sin(e+f*x)))],
[(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x)),x,7,-(a+b)*d*AppellF1(1/2,1/2,-1-m,3/2,1/2*(1-sin(e+f*x)),b*(1-sin(e+f*x))/(a+b))*cos(e+f*x)*(a+b*sin(e+f*x))^m*sqrt(2)/(b*f*((a+b*sin(e+f*x))/(a+b))^m*sqrt(1+sin(e+f*x)))-(b*c-a*d)*AppellF1(1/2,1/2,-m,3/2,1/2*(1-sin(e+f*x)),b*(1-sin(e+f*x))/(a+b))*cos(e+f*x)*(a+b*sin(e+f*x))^m*sqrt(2)/(b*f*((a+b*sin(e+f*x))/(a+b))^m*sqrt(1+sin(e+f*x)))],
[(a+b*sin(e+f*x))^m,x,3,-AppellF1(1/2,1/2,-m,3/2,1/2*(1-sin(e+f*x)),b*(1-sin(e+f*x))/(a+b))*cos(e+f*x)*(a+b*sin(e+f*x))^m*sqrt(2)/(f*((a+b*sin(e+f*x))/(a+b))^m*sqrt(1+sin(e+f*x)))],
[(a+b*sin(e+f*x))^m/(c+d*sin(e+f*x)),x,0,Unintegrable((a+b*sin(e+f*x))^m/(c+d*sin(e+f*x)),x)],
[(a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^2,x,0,Unintegrable((a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^2,x)],
[(a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^3,x,0,Unintegrable((a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^3,x)],
[(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(5/2),x,0,Unintegrable((a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(5/2),x)],
[(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(3/2),x,0,Unintegrable((a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(3/2),x)],
[(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(1/2),x,0,Unintegrable((a+b*sin(e+f*x))^m*sqrt(c+d*sin(e+f*x)),x)],
[(a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^(1/2),x,0,Unintegrable((a+b*sin(e+f*x))^m/sqrt(c+d*sin(e+f*x)),x)],
[(a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^(3/2),x,0,Unintegrable((a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^(3/2),x)],
[(a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^(5/2),x,0,Unintegrable((a+b*sin(e+f*x))^m/(c+d*sin(e+f*x))^(5/2),x)],

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

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

# Integrands of the form (a+a Sin[e+f x])^m (d Csc[e+f x])^n with n symbolic
[(d*csc(e+f*x))^n*(a+a*sin(e+f*x))^3,x,8,a^3*d^3*(1-2*n)*cot(e+f*x)*(d*csc(e+f*x))^(-3+n)/(f*(2-3*n+n^2))+d^3*cot(e+f*x)*(d*csc(e+f*x))^(-3+n)*(a^3+a^3*csc(e+f*x))/(f*(1-n))+a^3*d^3*(5-4*n)*cos(e+f*x)*(d*csc(e+f*x))^(-3+n)*hypergeom([1/2,1/2*(3-n)],[1/2*(5-n)],sin(e+f*x)^2)/(f*(3-4*n+n^2)*sqrt(cos(e+f*x)^2))+a^3*d^4*(11-4*n)*cos(e+f*x)*(d*csc(e+f*x))^(-4+n)*hypergeom([1/2,1/2*(4-n)],[1/2*(6-n)],sin(e+f*x)^2)/(f*(8-6*n+n^2)*sqrt(cos(e+f*x)^2))],
[(d*csc(e+f*x))^n*(a+a*sin(e+f*x))^2,x,7,a^2*d^2*cot(e+f*x)*(d*csc(e+f*x))^(-2+n)/(f*(1-n))+2*a^2*d^2*cos(e+f*x)*(d*csc(e+f*x))^(-2+n)*hypergeom([1/2,1/2*(2-n)],[1/2*(4-n)],sin(e+f*x)^2)/(f*(2-n)*sqrt(cos(e+f*x)^2))+a^2*d^3*(3-2*n)*cos(e+f*x)*(d*csc(e+f*x))^(-3+n)*hypergeom([1/2,1/2*(3-n)],[1/2*(5-n)],sin(e+f*x)^2)/(f*(3-4*n+n^2)*sqrt(cos(e+f*x)^2))],
[(d*csc(e+f*x))^n*(a+a*sin(e+f*x)),x,6,a*d*cos(e+f*x)*(d*csc(e+f*x))^(-1+n)*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],sin(e+f*x)^2)/(f*(1-n)*sqrt(cos(e+f*x)^2))+a*d^2*cos(e+f*x)*(d*csc(e+f*x))^(-2+n)*hypergeom([1/2,1/2*(2-n)],[1/2*(4-n)],sin(e+f*x)^2)/(f*(2-n)*sqrt(cos(e+f*x)^2))],
[(d*csc(e+f*x))^n/(a+a*sin(e+f*x)),x,7,-cot(e+f*x)*(d*csc(e+f*x))^n/(f*(a+a*csc(e+f*x)))+d*n*cos(e+f*x)*(d*csc(e+f*x))^(-1+n)*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],sin(e+f*x)^2)/(a*f*(1-n)*sqrt(cos(e+f*x)^2))+cos(e+f*x)*(d*csc(e+f*x))^n*hypergeom([1/2,-1/2*n],[1/2*(2-n)],sin(e+f*x)^2)/(a*f*sqrt(cos(e+f*x)^2))],
[(d*csc(e+f*x))^n/(a+a*sin(e+f*x))^2,x,8,-2/3*n*cot(e+f*x)*(d*csc(e+f*x))^(2+n)/(a^2*d^2*f*(1+csc(e+f*x)))+1/3*cot(e+f*x)*(d*csc(e+f*x))^(2+n)/(d^2*f*(a+a*csc(e+f*x))^2)+2/3*n*cos(e+f*x)*(d*csc(e+f*x))^(2+n)*hypergeom([1/2,1/2*(-2-n)],[-1/2*n],sin(e+f*x)^2)/(a^2*d^2*f*sqrt(cos(e+f*x)^2))-1/3*(1+2*n)*cos(e+f*x)*(d*csc(e+f*x))^(1+n)*hypergeom([1/2,1/2*(-1-n)],[1/2*(1-n)],sin(e+f*x)^2)/(a^2*d*f*sqrt(cos(e+f*x)^2))],

# Integrands of the form (a+a Sin[e+f x])^m (c (d Sin[e+f x])^p)^n with n and p symbolic
[(c*(d*sin(e+f*x))^p)^n*(a+a*sin(e+f*x))^m,x,5,-2^(1/2+m)*AppellF1(1/2,-n*p,1/2-m,3/2,1-sin(e+f*x),1/2*(1-sin(e+f*x)))*cos(e+f*x)*(c*(d*sin(e+f*x))^p)^n*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(f*sin(e+f*x)^(n*p))],
[(c*(d*sin(e+f*x))^p)^n*(a+a*sin(e+f*x))^3,x,7,-a^3*(7+2*n*p)*cos(e+f*x)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p)*(3+n*p))-cos(e+f*x)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n*(a^3+a^3*sin(e+f*x))/(f*(3+n*p))+a^3*(5+4*n*p)*cos(e+f*x)*hypergeom([1/2,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(1+n*p)*(2+n*p)*sqrt(cos(e+f*x)^2))+a^3*(11+4*n*p)*cos(e+f*x)*hypergeom([1/2,1/2*(2+n*p)],[1/2*(4+n*p)],sin(e+f*x)^2)*sin(e+f*x)^2*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p)*(3+n*p)*sqrt(cos(e+f*x)^2))],
[(c*(d*sin(e+f*x))^p)^n*(a+a*sin(e+f*x))^2,x,5,-a^2*cos(e+f*x)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p))+a^2*(3+2*n*p)*cos(e+f*x)*hypergeom([1/2,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(1+n*p)*(2+n*p)*sqrt(cos(e+f*x)^2))+2*a^2*cos(e+f*x)*hypergeom([1/2,1/2*(2+n*p)],[1/2*(4+n*p)],sin(e+f*x)^2)*sin(e+f*x)^2*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p)*sqrt(cos(e+f*x)^2))],
[(c*(d*sin(e+f*x))^p)^n*(a+a*sin(e+f*x)),x,4,a*cos(e+f*x)*hypergeom([1/2,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(1+n*p)*sqrt(cos(e+f*x)^2))+a*cos(e+f*x)*hypergeom([1/2,1/2*(2+n*p)],[1/2*(4+n*p)],sin(e+f*x)^2)*sin(e+f*x)^2*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p)*sqrt(cos(e+f*x)^2))],
[(c*(d*sin(e+f*x))^p)^n/(a+a*sin(e+f*x)),x,5,-cos(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(a+a*sin(e+f*x)))+cos(e+f*x)*hypergeom([1/2,1/2*n*p],[1/2*(2+n*p)],sin(e+f*x)^2)*(c*(d*sin(e+f*x))^p)^n/(a*f*sqrt(cos(e+f*x)^2))-n*p*cos(e+f*x)*hypergeom([1/2,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(a*f*(1+n*p)*sqrt(cos(e+f*x)^2))],
[(c*(d*sin(e+f*x))^p)^n/(a+a*sin(e+f*x))^2,x,6,2/3*(1-n*p)*cos(e+f*x)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(a^2*f*(1+sin(e+f*x)))+1/3*cos(e+f*x)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(a+a*sin(e+f*x))^2)-1/3*n*p*(1-2*n*p)*cos(e+f*x)*hypergeom([1/2,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(a^2*f*(1+n*p)*sqrt(cos(e+f*x)^2))+2/3*(1-n^2*p^2)*cos(e+f*x)*hypergeom([1/2,1/2*(2+n*p)],[1/2*(4+n*p)],sin(e+f*x)^2)*sin(e+f*x)^2*(c*(d*sin(e+f*x))^p)^n/(a^2*f*(2+n*p)*sqrt(cos(e+f*x)^2))],

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

# Integrands of the form (a+b Sin[e+f x])^m (d Csc[e+f x])^n with n symbolic
[(d*csc(e+f*x))^n*(a+b*sin(e+f*x))^3,x,8,a^2*b*d^3*(1-2*n)*cot(e+f*x)*(d*csc(e+f*x))^(-3+n)/(f*(2-3*n+n^2))+a^2*d^3*cot(e+f*x)*(d*csc(e+f*x))^(-3+n)*(b+a*csc(e+f*x))/(f*(1-n))+a*d^3*(3*b^2*(1-n)+a^2*(2-n))*cos(e+f*x)*(d*csc(e+f*x))^(-3+n)*hypergeom([1/2,1/2*(3-n)],[1/2*(5-n)],sin(e+f*x)^2)/(f*(3-4*n+n^2)*sqrt(cos(e+f*x)^2))+b*d^4*(b^2*(2-n)+3*a^2*(3-n))*cos(e+f*x)*(d*csc(e+f*x))^(-4+n)*hypergeom([1/2,1/2*(4-n)],[1/2*(6-n)],sin(e+f*x)^2)/(f*(8-6*n+n^2)*sqrt(cos(e+f*x)^2))],
[(d*csc(e+f*x))^n*(a+b*sin(e+f*x))^2,x,7,a^2*d^2*cot(e+f*x)*(d*csc(e+f*x))^(-2+n)/(f*(1-n))+2*a*b*d^2*cos(e+f*x)*(d*csc(e+f*x))^(-2+n)*hypergeom([1/2,1/2*(2-n)],[1/2*(4-n)],sin(e+f*x)^2)/(f*(2-n)*sqrt(cos(e+f*x)^2))+d^3*(b^2*(1-n)+a^2*(2-n))*cos(e+f*x)*(d*csc(e+f*x))^(-3+n)*hypergeom([1/2,1/2*(3-n)],[1/2*(5-n)],sin(e+f*x)^2)/(f*(3-4*n+n^2)*sqrt(cos(e+f*x)^2))],
[(d*csc(e+f*x))^n*(a+b*sin(e+f*x)),x,6,a*d*cos(e+f*x)*(d*csc(e+f*x))^(-1+n)*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],sin(e+f*x)^2)/(f*(1-n)*sqrt(cos(e+f*x)^2))+b*d^2*cos(e+f*x)*(d*csc(e+f*x))^(-2+n)*hypergeom([1/2,1/2*(2-n)],[1/2*(4-n)],sin(e+f*x)^2)/(f*(2-n)*sqrt(cos(e+f*x)^2))],
[(d*csc(e+f*x))^n/(a+b*sin(e+f*x)),x,7,b*AppellF1(1/2,1/2*n,1,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(1+n)*sin(e+f*x)*(sin(e+f*x)^2)^(1/2*n)/((a^2-b^2)*d*f)-a*AppellF1(1/2,1/2*(1+n),1,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(1+n)*(sin(e+f*x)^2)^(1/2*(1+n))/((a^2-b^2)*d*f)],
[(d*csc(e+f*x))^n/(a+b*sin(e+f*x))^2,x,10,-b^2*AppellF1(1/2,1/2*(-1+n),2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(2+n)*sin(e+f*x)^3*(sin(e+f*x)^2)^(1/2*(-1+n))/((a^2-b^2)^2*d^2*f)-a^2*AppellF1(1/2,1/2*(1+n),2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(2+n)*sin(e+f*x)*(sin(e+f*x)^2)^(1/2*(1+n))/((a^2-b^2)^2*d^2*f)+2*a*b*AppellF1(1/2,1/2*n,2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(2+n)*(sin(e+f*x)^2)^(1/2*(2+n))/((a^2-b^2)^2*d^2*f)],
[(d*csc(e+f*x))^n/(a+b*sin(e+f*x))^3,x,12,-3*a*b^2*AppellF1(1/2,1/2*(-1+n),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(3+n)*sin(e+f*x)^4*(sin(e+f*x)^2)^(1/2*(-1+n))/((a^2-b^2)^3*d^3*f)+b^3*AppellF1(1/2,1/2*(-2+n),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(3+n)*sin(e+f*x)^3*(sin(e+f*x)^2)^(1/2*n)/((a^2-b^2)^3*d^3*f)+3*a^2*b*AppellF1(1/2,1/2*n,3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(3+n)*sin(e+f*x)^3*(sin(e+f*x)^2)^(1/2*n)/((a^2-b^2)^3*d^3*f)-a^3*AppellF1(1/2,1/2*(1+n),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*csc(e+f*x))^(3+n)*(sin(e+f*x)^2)^(1/2*(3+n))/((a^2-b^2)^3*d^3*f)],

# Integrands of the form (a+b Sin[e+f x])^m (c (d Sin[e+f x])^p)^n with n and p symbolic
[(c*(d*sin(e+f*x))^p)^n*(a+b*sin(e+f*x))^m,x,1,(c*(d*sin(e+f*x))^p)^n*Unintegrable((d*sin(e+f*x))^(n*p)*(a+b*sin(e+f*x))^m,x)/(d*sin(e+f*x))^(n*p)],
[(c*(d*sin(e+f*x))^p)^n*(a+b*sin(e+f*x))^3,x,6,-a*b^2*(7+2*n*p)*cos(e+f*x)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p)*(3+n*p))-b^2*cos(e+f*x)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n*(a+b*sin(e+f*x))/(f*(3+n*p))+a*(3*b^2*(1+n*p)+a^2*(2+n*p))*cos(e+f*x)*hypergeom([1/2,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(1+n*p)*(2+n*p)*sqrt(cos(e+f*x)^2))+b*(b^2*(2+n*p)+3*a^2*(3+n*p))*cos(e+f*x)*hypergeom([1/2,1/2*(2+n*p)],[1/2*(4+n*p)],sin(e+f*x)^2)*sin(e+f*x)^2*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p)*(3+n*p)*sqrt(cos(e+f*x)^2))],
[(c*(d*sin(e+f*x))^p)^n*(a+b*sin(e+f*x))^2,x,5,-b^2*cos(e+f*x)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p))+(b^2*(1+n*p)+a^2*(2+n*p))*cos(e+f*x)*hypergeom([1/2,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(1+n*p)*(2+n*p)*sqrt(cos(e+f*x)^2))+2*a*b*cos(e+f*x)*hypergeom([1/2,1/2*(2+n*p)],[1/2*(4+n*p)],sin(e+f*x)^2)*sin(e+f*x)^2*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p)*sqrt(cos(e+f*x)^2))],
[(c*(d*sin(e+f*x))^p)^n*(a+b*sin(e+f*x)),x,4,a*cos(e+f*x)*hypergeom([1/2,1/2*(1+n*p)],[1/2*(3+n*p)],sin(e+f*x)^2)*sin(e+f*x)*(c*(d*sin(e+f*x))^p)^n/(f*(1+n*p)*sqrt(cos(e+f*x)^2))+b*cos(e+f*x)*hypergeom([1/2,1/2*(2+n*p)],[1/2*(4+n*p)],sin(e+f*x)^2)*sin(e+f*x)^2*(c*(d*sin(e+f*x))^p)^n/(f*(2+n*p)*sqrt(cos(e+f*x)^2))],
[(c*(d*sin(e+f*x))^p)^n/(a+b*sin(e+f*x)),x,6,b*AppellF1(1/2,-1/2*n*p,1,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)*f*(sin(e+f*x)^2)^(1/2*n*p))-a*AppellF1(1/2,1/2*(1-n*p),1,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cot(e+f*x)*(sin(e+f*x)^2)^(1/2*(1-n*p))*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)*f)],
[(c*(d*sin(e+f*x))^p)^n/(a+b*sin(e+f*x))^2,x,11,2*a*b*AppellF1(1/2,-1/2*n*p,2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)^2*f*(sin(e+f*x)^2)^(1/2*n*p))-b^2*AppellF1(1/2,1/2*(-1-n*p),2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*sin(e+f*x)*(sin(e+f*x)^2)^(1/2*(-1-n*p))*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)^2*f)-a^2*AppellF1(1/2,1/2*(1-n*p),2,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cot(e+f*x)*(sin(e+f*x)^2)^(1/2*(1-n*p))*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)^2*f)],
[(c*(d*sin(e+f*x))^p)^n/(a+b*sin(e+f*x))^3,x,14,3*a^2*b*AppellF1(1/2,-1/2*n*p,3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)^3*f*(sin(e+f*x)^2)^(1/2*n*p))+b^3*AppellF1(1/2,1/2*(-2-n*p),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)^3*f*(sin(e+f*x)^2)^(1/2*n*p))-3*a*b^2*AppellF1(1/2,1/2*(-1-n*p),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*sin(e+f*x)*(sin(e+f*x)^2)^(1/2*(-1-n*p))*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)^3*f)-a^3*AppellF1(1/2,1/2*(1-n*p),3,3/2,cos(e+f*x)^2,-b^2*cos(e+f*x)^2/(a^2-b^2))*cot(e+f*x)*(sin(e+f*x)^2)^(1/2*(1-n*p))*(c*(d*sin(e+f*x))^p)^n/((a^2-b^2)^3*f)]]:
