\[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {3 e x}{2 a}+\frac {3 f \,x^{2}}{4 a}+\frac {\left (f x +e \right ) \cos \left (d x +c \right )}{a d}+\frac {\left (f x +e \right ) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \ln \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a \,d^{2}}-\frac {f \sin \left (d x +c \right )}{a \,d^{2}}-\frac {\left (f x +e \right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 a d}+\frac {f \left (\sin ^{2}\left (d x +c \right )\right )}{4 a \,d^{2}} \]
command
integrate((f*x+e)*sin(d*x+c)^3/(a+a*sin(d*x+c)),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 \[ \text {Timed out} \]_______________________________________________________