\[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {63 \cos \left (f x +e \right )}{128 a^{3} c f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {21 \sec \left (f x +e \right )}{80 a^{3} c f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {63 \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {c}\, \sqrt {2}}{2 \sqrt {c -c \sin \left (f x +e \right )}}\right ) \sqrt {2}}{256 a^{3} c^{\frac {5}{2}} f}-\frac {21 \sec \left (f x +e \right )}{32 a^{3} c^{2} f \sqrt {c -c \sin \left (f x +e \right )}}-\frac {3 \left (\sec ^{3}\left (f x +e \right )\right )}{10 a^{3} c^{2} f \sqrt {c -c \sin \left (f x +e \right )}}-\frac {\left (\sec ^{5}\left (f x +e \right )\right ) \sqrt {c -c \sin \left (f x +e \right )}}{5 a^{3} c^{3} f} \]
command
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(5/2),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 \[ \text {Exception raised: TypeError} \]_______________________________________________________