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