\[ \int \csc (e+f x) (b \sec (e+f x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{f}-\frac {b^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{f}+\frac {2 b \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f} \]
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
\[ \frac {b^{6} {\left (\frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} - \frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {7}{2}}} + \frac {2}{\sqrt {b \cos \left (f x + e\right )} b^{3} \cos \left (f x + e\right )}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________