\[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^2} \, dx \]
Optimal antiderivative \[ -\frac {32 \left (5 A -11 B \right ) c^{2} \sec \left (f x +e \right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{15 a^{2} f}-\frac {4 \left (5 A -11 B \right ) c \sec \left (f x +e \right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{15 a^{2} f}-\frac {\left (5 A -11 B \right ) \sec \left (f x +e \right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{15 a^{2} f}-\frac {\left (A -B \right ) \left (\sec ^{3}\left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {11}{2}}}{3 a^{2} c^{2} f}+\frac {128 \left (5 A -11 B \right ) c^{3} \sec \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}}{15 a^{2} f} \]
command
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________