\[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (7 A +3 B \right ) \cos \left (f x +e \right )}{16 a^{3} f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\left (A -B \right ) \left (\sec ^{5}\left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{5 a^{3} c^{3} f}+\frac {\left (7 A +3 B \right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {c}\, \sqrt {2}}{2 \sqrt {c -c \sin \left (f x +e \right )}}\right ) \sqrt {2}}{32 a^{3} c^{\frac {3}{2}} f}-\frac {\left (7 A +3 B \right ) \sec \left (f x +e \right )}{12 a^{3} c f \sqrt {c -c \sin \left (f x +e \right )}}-\frac {\left (7 A +3 B \right ) \left (\sec ^{3}\left (f x +e \right )\right ) \sqrt {c -c \sin \left (f x +e \right )}}{30 a^{3} c^{2} f} \]
command
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(3/2),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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________