\[ \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {3 f^{3} x}{8 a \,d^{3}}+\frac {\left (f x +e \right )^{3}}{4 a d}-\frac {6 f^{3} \cos \left (d x +c \right )}{a \,d^{4}}+\frac {3 f \left (f x +e \right )^{2} \cos \left (d x +c \right )}{a \,d^{2}}-\frac {6 f^{2} \left (f x +e \right ) \sin \left (d x +c \right )}{a \,d^{3}}+\frac {\left (f x +e \right )^{3} \sin \left (d x +c \right )}{a d}+\frac {3 f^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 a \,d^{4}}-\frac {3 f \left (f x +e \right )^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{4 a \,d^{2}}+\frac {3 f^{2} \left (f x +e \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{4 a \,d^{3}}-\frac {\left (f x +e \right )^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2 a d} \]
command
integrate((f*x+e)^3*cos(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 \[ \text {Timed out} \]_______________________________________________________