\[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {3 \arctan \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{4 b^{\frac {5}{2}} f}+\frac {3 \arctanh \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{4 b^{\frac {5}{2}} f}-\frac {\left (\cot ^{2}\left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{2 b^{3} f} \]
command
integrate(csc(f*x+e)^3/(b*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {2 \, \sqrt {b \cos \left (f x + e\right )} b \cos \left (f x + e\right )}{b^{2} \cos \left (f x + e\right )^{2} - b^{2}} - \frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{\sqrt {b}}}{4 \, b^{2} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________