\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {d \left (2 A \left (3 c -2 d \right ) d +B \left (6 c^{2}-12 c d +7 d^{2}\right )\right ) x}{2 a^{2}}+\frac {2 d \left (A \left (c^{2}+6 c d -5 d^{2}\right )+B \left (2 c^{2}-15 c d +8 d^{2}\right )\right ) \cos \left (f x +e \right )}{3 a^{2} f}+\frac {d^{2} \left (B \left (4 c -21 d \right )+2 A \left (c +6 d \right )\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right )}{6 a^{2} f}-\frac {\left (2 B \left (c -4 d \right )+A \left (c +5 d \right )\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{2}}{3 a^{2} f \left (1+\sin \left (f x +e \right )\right )}-\frac {\left (A -B \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{3 f \left (a +a \sin \left (f x +e \right )\right )^{2}} \]
command
integrate((A+B*sin(f*x+e))*(c+d*sin(f*x+e))**3/(a+a*sin(f*x+e))**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________