\[ \int \csc ^5(e+f x) \sqrt {b \sec (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {7 \left (\cot ^{2}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{16 b f}-\frac {\left (\cot ^{4}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{\frac {7}{2}}}{4 b^{3} f}+\frac {21 \arctan \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right ) \sqrt {b}}{32 f}-\frac {21 \arctanh \left (\frac {\sqrt {b \sec \left (f x +e \right )}}{\sqrt {b}}\right ) \sqrt {b}}{32 f} \]
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
\[ \frac {b^{6} {\left (\frac {21 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{5}} - \frac {21 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {11}{2}}} + \frac {2 \, {\left (7 \, \sqrt {b \cos \left (f x + e\right )} b^{2} \cos \left (f x + e\right )^{2} - 11 \, \sqrt {b \cos \left (f x + e\right )} b^{2}\right )}}{{\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )}^{2} b^{4}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{32 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________