\[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\cos \left (f x +e \right )}{4 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {3 \cos \left (f x +e \right )}{8 a f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {3 \cos \left (f x +e \right )}{8 a^{2} f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a +a \sin \left (f x +e \right )}}+\frac {3 \arctanh \left (\sin \left (f x +e \right )\right ) \cos \left (f x +e \right )}{8 a^{2} c f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}} \]
command
integrate(1/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {c} {\left (\frac {6 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {12 \, \log \left ({\left | \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right )}{a^{\frac {5}{2}} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {6 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{\frac {5}{2}} c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{32 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________