# Maple integration test file: "4 Trig functions\4.1 Sine\4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).txt"

lst:=[

# Integrands of the form (b Sin[e+f x])^m (A+C Sin[e+f x]^2)

# Integrands of the form (b Sin[e+f x])^m (A+C Sin[e+f x]^2) when A (m+2)+C (m+1)=0
[sin(e+f*x)^m*(1+m-(2+m)*sin(e+f*x)^2),x,1,cos(e+f*x)*sin(e+f*x)^(1+m)/f],
[sin(e+f*x)^5*(6-7*sin(e+f*x)^2),x,1,cos(e+f*x)*sin(e+f*x)^6/f],
[sin(e+f*x)^4*(5-6*sin(e+f*x)^2),x,1,cos(e+f*x)*sin(e+f*x)^5/f],
[sin(e+f*x)^3*(4-5*sin(e+f*x)^2),x,1,cos(e+f*x)*sin(e+f*x)^4/f],
[sin(e+f*x)^2*(3-4*sin(e+f*x)^2),x,1,cos(e+f*x)*sin(e+f*x)^3/f],
[sin(e+f*x)*(2-3*sin(e+f*x)^2),x,1,cos(e+f*x)*sin(e+f*x)^2/f],
[1-2*sin(e+f*x)^2,x,3,cos(e+f*x)*sin(e+f*x)/f],
[-csc(e+f*x)*sin(e+f*x)^2,x,1,cos(e+f*x)/f],
[-csc(e+f*x)^2,x,2,cot(e+f*x)/f],
[csc(e+f*x)^3*(-2+sin(e+f*x)^2),x,1,cot(e+f*x)*csc(e+f*x)/f],
[csc(e+f*x)^4*(-3+2*sin(e+f*x)^2),x,1,cot(e+f*x)*csc(e+f*x)^2/f],
[csc(e+f*x)^5*(-4+3*sin(e+f*x)^2),x,1,cot(e+f*x)*csc(e+f*x)^3/f],

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

# Integrands of the form (a+a Sin[e+f x])^m (A+C Sin[e+f x]^2)
[(a+a*sin(e+f*x))^m*(A+C*sin(e+f*x)^2),x,4,C*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(2+3*m+m^2))-2^(1/2+m)*(C*(1+m+m^2)+A*(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))-C*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(2+m))],

# Integrands of the form (a+b Sin[e+f x])^m (A+C Sin[e+f x]^2)
[(a+b*sin(e+f*x))^m*(A-A*sin(e+f*x)^2),x,7,4*A*AppellF1(1/2,-3/2,-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)))-4*A*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*(A+C*sin(e+f*x)^2),x,8,-C*cos(e+f*x)*(a+b*sin(e+f*x))^(1+m)/(b*f*(2+m))+a*(a+b)*C*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^2*C+b^2*(C*(1+m)+A*(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)))],
[sin(e+f*x)^5*(A+C*sin(e+f*x)^2),x,3,-(A+C)*cos(e+f*x)/f+1/3*(2*A+3*C)*cos(e+f*x)^3/f-1/5*(A+3*C)*cos(e+f*x)^5/f+1/7*C*cos(e+f*x)^7/f],

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

# Integrands of the form (a+a Sin[e+f x])^m (A+B Sin[e+f x]+C Sin[e+f x]^2)
[(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)+C*sin(e+f*x)^2),x,4,(C-B*(2+m))*cos(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1+m)*(2+m))-2^(1/2+m)*(B*m*(2+m)+C*(1+m+m^2)+A*(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))-C*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(2+m))],

# Integrands of the form (a+b Sin[e+f x])^m (A+B Sin[e+f x]+C Sin[e+f x]^2)
[(a+b*sin(e+f*x))^m*(A+(A+C)*sin(e+f*x)+C*sin(e+f*x)^2),x,7,-4*C*AppellF1(1/2,-3/2,-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)))-2*(A-C)*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*(A+B*sin(e+f*x)+C*sin(e+f*x)^2),x,8,-C*cos(e+f*x)*(a+b*sin(e+f*x))^(1+m)/(b*f*(2+m))+(a+b)*(a*C-b*B*(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^2*C+b^2*C*(1+m)+A*b^2*(2+m)-a*b*B*(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)))]]:
