\[ \int \csc ^3(e+f x) (b \sec (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {5 b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{4 f}-\frac {5 b^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{4 f}-\frac {\left (\cot ^{2}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}}{2 b f}+\frac {5 b \sqrt {b \sec \left (f x +e \right )}}{2 f} \]
command
integrate(csc(f*x+e)^3*(b*sec(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {b^{6} {\left (\frac {5 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{4}} + \frac {5 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {9}{2}}} + \frac {2 \, {\left (5 \, b^{2} \cos \left (f x + e\right )^{2} - 4 \, b^{2}\right )}}{{\left (\sqrt {b \cos \left (f x + e\right )} b^{2} \cos \left (f x + e\right )^{2} - \sqrt {b \cos \left (f x + e\right )} b^{2}\right )} b^{4}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{4 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________