\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {a^{3} \left (A +B \right ) c^{3} \left (\cos ^{7}\left (f x +e \right )\right )}{4 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {11}{2}}}-\frac {a^{3} \left (3 A +11 B \right ) c \left (\cos ^{5}\left (f x +e \right )\right )}{8 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}-\frac {5 a^{3} \left (3 A +11 B \right ) \left (\cos ^{3}\left (f x +e \right )\right )}{24 c f \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {5 a^{3} \left (3 A +11 B \right ) \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {c}\, \sqrt {2}}{2 \sqrt {c -c \sin \left (f x +e \right )}}\right ) \sqrt {2}}{4 c^{\frac {5}{2}} f}-\frac {5 a^{3} \left (3 A +11 B \right ) \cos \left (f x +e \right )}{4 c^{2} f \sqrt {c -c \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________