\[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {\left (A +B \right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {c}\, \sqrt {2}}{2 \sqrt {c -c \sin \left (f x +e \right )}}\right ) \sqrt {2}}{2 a f \sqrt {c}}-\frac {\left (A -B \right ) \sec \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}}{a c f} \]
command
integrate((A+B*sin(f*x+e))/(a+a*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
\[ \frac {\frac {\sqrt {2} {\left (A \sqrt {c} + B \sqrt {c}\right )} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{a c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {4 \, \sqrt {2} {\left (A \sqrt {c} - B \sqrt {c}\right )}}{a c {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{4 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________