\[ \int \frac {(c+d \sin (e+f x))^2}{\sqrt {a+a \sin (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (c -d \right )^{2} \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{f \sqrt {a}}-\frac {4 \left (3 c -d \right ) d \cos \left (f x +e \right )}{3 f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 d^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{3 a f} \]
command
integrate((c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3 \, \sqrt {2} {\left (\sqrt {a} c^{2} - 2 \, \sqrt {a} c d + \sqrt {a} d^{2}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (\sqrt {a} c^{2} - 2 \, \sqrt {a} c d + \sqrt {a} d^{2}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {8 \, \sqrt {2} {\left (a^{\frac {5}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{\frac {5}{2}} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________