\[ \int \frac {\cos ^8(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {3 x}{256 a}+\frac {\cos ^{7}\left (d x +c \right )}{7 a d}-\frac {2 \left (\cos ^{9}\left (d x +c \right )\right )}{9 a d}+\frac {\cos ^{11}\left (d x +c \right )}{11 a d}+\frac {3 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{256 a d}+\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{128 a d}+\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{160 a d}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80 a d}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{10 a d} \]
command
integrate(cos(d*x+c)**8*sin(d*x+c)**4/(a+a*sin(d*x+c)),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 \[ \text {Timed out} \]_____________________________________________________