\[ \int \frac {\cos ^8(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx \]
Optimal antiderivative \[ -\frac {7 x}{16 a^{3}}-\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{30 a^{3} d}-\frac {7 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{16 a^{3} d}-\frac {7 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{24 a^{3} d}-\frac {\cos ^{9}\left (d x +c \right )}{3 d \left (a +a \sin \left (d x +c \right )\right )^{3}}-\frac {\cos ^{7}\left (d x +c \right )}{6 d \left (a^{3}+a^{3} \sin \left (d x +c \right )\right )} \]
command
integrate(cos(d*x+c)**8*sin(d*x+c)/(a+a*sin(d*x+c))**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 \[ \text {Timed out} \]_____________________________________________________