\[ \int \frac {1}{(a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))} \, dx \]
Optimal antiderivative \[ -\frac {\cos \left (f x +e \right )}{4 \left (c -d \right ) f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {\left (3 c -11 d \right ) \cos \left (f x +e \right )}{16 a \left (c -d \right )^{2} f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\left (3 c^{2}-14 c d +43 d^{2}\right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{32 a^{\frac {5}{2}} \left (c -d \right )^{3} f}+\frac {2 d^{\frac {5}{2}} \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {d}}{\sqrt {c +d}\, \sqrt {a +a \sin \left (f x +e \right )}}\right )}{a^{\frac {5}{2}} \left (c -d \right )^{3} f \sqrt {c +d}} \]
command
integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e)),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 \[ \text {Exception raised: TypeError} \]_______________________________________________________