\[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {a \cos \left (f x +e \right ) \ln \left (1-\sin \left (f x +e \right )\right )}{f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {\ln \left (\sin \left (f x +e \right )\right ) \sec \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}{c f} \]
command
integrate((a+a*sin(f*x+e))^(1/2)/sin(f*x+e)/(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ 0 \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________