\[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx \]
Optimal antiderivative \[ -\frac {23 x}{16 a^{3}}-\frac {4 \cos \left (d x +c \right )}{a^{3} d}+\frac {7 \left (\cos ^{3}\left (d x +c \right )\right )}{3 a^{3} d}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{5 a^{3} d}+\frac {23 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{16 a^{3} d}+\frac {23 \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{24 a^{3} d}+\frac {\cos \left (d x +c \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{6 a^{3} d} \]
command
integrate(cos(d*x+c)**6*sin(d*x+c)**3/(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 \[ \text {Timed out} \]_____________________________________________________