\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {a^{3} \left (A +B \right ) c^{3} \left (\cos ^{7}\left (f x +e \right )\right )}{2 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}}}+\frac {a^{3} \left (5 A +9 B \right ) c \left (\cos ^{5}\left (f x +e \right )\right )}{10 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {a^{3} \left (5 A +9 B \right ) \left (\cos ^{3}\left (f x +e \right )\right )}{3 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2 a^{3} \left (5 A +9 B \right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {c}\, \sqrt {2}}{2 \sqrt {c -c \sin \left (f x +e \right )}}\right ) \sqrt {2}}{c^{\frac {3}{2}} f}+\frac {2 a^{3} \left (5 A +9 B \right ) \cos \left (f x +e \right )}{c f \sqrt {c -c \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(3/2),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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________