\[ \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^2} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {d}}{\sqrt {c +d}\, \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {a}}{\left (c +d \right )^{\frac {3}{2}} f \sqrt {d}}-\frac {a \cos \left (f x +e \right )}{\left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right ) \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} \sqrt {a} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c d - d^{2}}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\sqrt {-c d - d^{2}} {\left (c + d\right )}} + \frac {2 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (2 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )} {\left (c + d\right )}}\right )}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________