\[ \int \frac {(a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right )}{\sqrt {c-c \sin (e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {\left (A +C \right ) \cos \left (f x +e \right ) \hypergeom \left (\left [1, \frac {1}{2}+m \right ], \left [\frac {3}{2}+m \right ], \frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{f \left (1+2 m \right ) \sqrt {c -c \sin \left (f x +e \right )}}-\frac {2 C \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{1+m}}{a f \left (3+2 m \right ) \sqrt {c -c \sin \left (f x +e \right )}} \]
command
Integrate[((a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2))/Sqrt[c - c*Sin[e + f*x]],x]
Mathematica 13.1 output
\[ \text {Result too large to show} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________