\[ \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {3 \arctan \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{32 b^{\frac {3}{2}} f}+\frac {3 \arctanh \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{32 b^{\frac {3}{2}} f}-\frac {3 \left (\cot ^{2}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{16 b^{3} f}-\frac {\left (\cot ^{4}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{4 b^{3} f} \]
command
integrate(csc(f*x+e)^5/(b*sec(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {b^{2} {\left (\frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} - \frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {7}{2}}} + \frac {2 \, {\left (\sqrt {b \cos \left (f x + e\right )} b^{2} \cos \left (f x + e\right )^{2} + 3 \, \sqrt {b \cos \left (f x + e\right )} b^{2}\right )}}{{\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )}^{2} b^{2}}\right )}}{32 \, f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________