\[ \int \frac {\csc ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (\cot ^{4}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}}{4 b^{3} f}-\frac {5 \arctan \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{32 f \sqrt {b}}-\frac {5 \arctanh \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right )}{32 f \sqrt {b}}-\frac {5 \left (\cot ^{2}\left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{16 b f} \]
command
integrate(csc(f*x+e)^5/(b*sec(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {b^{4} {\left (\frac {5 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{4}} + \frac {5 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {9}{2}}} + \frac {2 \, {\left (5 \, \sqrt {b \cos \left (f x + e\right )} b^{3} \cos \left (f x + e\right )^{3} - 9 \, \sqrt {b \cos \left (f x + e\right )} b^{3} \cos \left (f x + e\right )\right )}}{{\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )}^{2} b^{4}}\right )}}{32 \, f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________