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