\[ \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {5 \left (2 c -d \right ) d^{2} \left (2 c^{2}-3 c d +2 d^{2}\right ) x}{2 a^{2}}+\frac {2 d \left (c^{4}+10 c^{3} d -44 c^{2} d^{2}+40 c \,d^{3}-12 d^{4}\right ) \cos \left (f x +e \right )}{3 a^{2} f}+\frac {d^{2} \left (2 c^{3}+20 c^{2} d -57 c \,d^{2}+30 d^{3}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right )}{6 a^{2} f}+\frac {d \left (c^{2}+10 c d -12 d^{2}\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{2}}{3 a^{2} f}-\frac {\left (c -d \right ) \left (c +10 d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{3 a^{2} f \left (1+\sin \left (f x +e \right )\right )}-\frac {\left (c -d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{4}}{3 f \left (a +a \sin \left (f x +e \right )\right )^{2}} \]
command
integrate((c+d*sin(f*x+e))**5/(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 \[ \text {Timed out} \]_____________________________________________________