\[ \int \frac {\cos ^4(c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx \]
Optimal antiderivative \[ -\frac {5 x}{8 a^{2}}-\frac {2 \cos \left (d x +c \right )}{a^{2} d}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{3 a^{2} d}-\frac {4 \left (\cos ^{5}\left (d x +c \right )\right )}{5 a^{2} d}+\frac {\cos ^{7}\left (d x +c \right )}{7 a^{2} d}+\frac {5 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 a^{2} d}+\frac {5 \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{12 a^{2} d}+\frac {\cos \left (d x +c \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{3 a^{2} d} \]
command
integrate(cos(d*x+c)**4*sin(d*x+c)**5/(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 \[ \text {Timed out} \]_____________________________________________________