\[ \int \frac {\cos ^8(c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {11 x}{128 a^{2}}+\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{5 a^{2} d}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{7 a^{2} d}+\frac {11 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{128 a^{2} d}+\frac {11 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{192 a^{2} d}-\frac {11 \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{48 a^{2} d}-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{8 a^{2} d} \]
command
integrate(cos(d*x+c)**8*sin(d*x+c)**2/(a+a*sin(d*x+c))**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 \[ \text {Timed out} \]_____________________________________________________