\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^2}{\sqrt {a+a \sin (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \left (c -d \right )^{2} \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{f \sqrt {a}}-\frac {4 \left (5 A \left (3 c -d \right ) d +B \left (6 c^{2}-7 c d +7 d^{2}\right )\right ) \cos \left (f x +e \right )}{15 f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 B \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{2}}{5 f \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 d \left (5 A d +4 B c -B d \right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{15 a f} \]
command
integrate((A+B*sin(f*x+e))*(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
\[ \frac {\frac {15 \, \sqrt {2} {\left (A \sqrt {a} c^{2} - B \sqrt {a} c^{2} - 2 \, A \sqrt {a} c d + 2 \, B \sqrt {a} c d + A \sqrt {a} d^{2} - B \sqrt {a} d^{2}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {15 \, \sqrt {2} {\left (A \sqrt {a} c^{2} - B \sqrt {a} c^{2} - 2 \, A \sqrt {a} c d + 2 \, B \sqrt {a} c d + A \sqrt {a} d^{2} - B \sqrt {a} d^{2}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {4 \, \sqrt {2} {\left (12 \, B a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 20 \, B a^{\frac {9}{2}} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, A a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, B a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, B a^{\frac {9}{2}} c^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, A a^{\frac {9}{2}} c d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, B a^{\frac {9}{2}} d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{5} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{30 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________