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