\[ \int \frac {(c+d \sin (e+f x))^6}{(a+a \sin (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {d^{3} \left (40 c^{3}-90 c^{2} d +78 c \,d^{2}-23 d^{3}\right ) x}{2 a^{3}}+\frac {2 d \left (2 c^{5}+18 c^{4} d +107 c^{3} d^{2}-472 c^{2} d^{3}+456 c \,d^{4}-136 d^{5}\right ) \cos \left (f x +e \right )}{15 a^{3} f}+\frac {d^{2} \left (4 c^{4}+36 c^{3} d +216 c^{2} d^{2}-626 c \,d^{3}+345 d^{4}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right )}{30 a^{3} f}+\frac {d \left (2 c^{3}+18 c^{2} d +111 c \,d^{2}-136 d^{3}\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{2}}{15 a^{3} f}-\frac {\left (c -d \right ) \left (2 c^{2}+18 c d +115 d^{2}\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{3}}{15 f \left (a^{3}+a^{3} \sin \left (f x +e \right )\right )}-\frac {\left (c -d \right ) \left (2 c +13 d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{4}}{15 a f \left (a +a \sin \left (f x +e \right )\right )^{2}}-\frac {\left (c -d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{5}}{5 f \left (a +a \sin \left (f x +e \right )\right )^{3}} \]
command
integrate((c+d*sin(f*x+e))**6/(a+a*sin(f*x+e))**3,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 \[ \text {Timed out} \]_____________________________________________________