\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{2 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\left (c -d \right )^{2} \left (3 B \left (c -5 d \right )+A \left (c +11 d \right )\right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} f}+\frac {d \left (15 A \,c^{2}-120 A c d +65 A \,d^{2}-99 B \,c^{2}+168 B c d -93 B \,d^{2}\right ) \cos \left (f x +e \right )}{15 a f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {\left (5 A -9 B \right ) d \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{2}}{10 a f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {d^{2} \left (15 A c -35 A d -51 B c +39 B d \right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{30 a^{2} f} \]
command
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {15 \, \sqrt {2} {\left (A \sqrt {a} c^{3} + 3 \, B \sqrt {a} c^{3} + 9 \, A \sqrt {a} c^{2} d - 21 \, B \sqrt {a} c^{2} d - 21 \, A \sqrt {a} c d^{2} + 33 \, B \sqrt {a} c d^{2} + 11 \, A \sqrt {a} d^{3} - 15 \, B \sqrt {a} d^{3}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \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^{3} + 3 \, B \sqrt {a} c^{3} + 9 \, A \sqrt {a} c^{2} d - 21 \, B \sqrt {a} c^{2} d - 21 \, A \sqrt {a} c d^{2} + 33 \, B \sqrt {a} c d^{2} + 11 \, A \sqrt {a} d^{3} - 15 \, B \sqrt {a} d^{3}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {30 \, \sqrt {2} {\left (A \sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - B \sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, A \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, B \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {16 \, \sqrt {2} {\left (12 \, B a^{\frac {17}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 30 \, B a^{\frac {17}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 \, A a^{\frac {17}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45 \, B a^{\frac {17}{2}} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 45 \, A a^{\frac {17}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 45 \, B a^{\frac {17}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, A a^{\frac {17}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, B a^{\frac {17}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{10} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{120 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________