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