\[ \int \csc ^3(e+f x) (b \sec (e+f x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {7 b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{4 f}-\frac {7 b^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{4 f}+\frac {7 b \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{6 f}-\frac {\left (\cot ^{2}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{\frac {7}{2}}}{2 b f} \]
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
\[ \frac {b^{8} {\left (\frac {6 \, \sqrt {b \cos \left (f x + e\right )}}{{\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )} b^{4}} + \frac {21 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{5}} - \frac {21 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {11}{2}}} + \frac {8}{\sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{12 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________