\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (3 A c +9 A d +5 B c -17 B d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{2}}{16 a f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\left (A -B \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{4 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {\left (c -d \right ) \left (B \left (5 c^{2}+62 c d -163 d^{2}\right )+3 A \left (c^{2}+6 c d +25 d^{2}\right )\right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{32 a^{\frac {5}{2}} f}+\frac {d \left (A \left (9 c^{2}+36 c d -93 d^{2}\right )+B \left (15 c^{2}-228 c d +197 d^{2}\right )\right ) \cos \left (f x +e \right )}{24 a^{2} f \sqrt {a +a \sin \left (f x +e \right )}}+\frac {d^{2} \left (9 A c +39 A d +15 B c -95 B d \right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}}{48 a^{3} f} \]
command
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________