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