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