\[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx \]
Optimal antiderivative \[ -\frac {\cos ^{3}\left (d x +c \right )}{13 d \left (a +a \sin \left (d x +c \right )\right )^{8}}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{143 a d \left (a +a \sin \left (d x +c \right )\right )^{7}}-\frac {20 \left (\cos ^{3}\left (d x +c \right )\right )}{1287 a^{2} d \left (a +a \sin \left (d x +c \right )\right )^{6}}-\frac {20 \left (\cos ^{3}\left (d x +c \right )\right )}{3003 a^{3} d \left (a +a \sin \left (d x +c \right )\right )^{5}}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{3003 d \left (a^{2}+a^{2} \sin \left (d x +c \right )\right )^{4}}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{9009 a^{2} d \left (a^{2}+a^{2} \sin \left (d x +c \right )\right )^{3}} \]
command
integrate(cos(d*x+c)**2/(a+a*sin(d*x+c))**8,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 \[ \text {Timed out} \]_____________________________________________________