\[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {a \left (\sin ^{1+n}\left (d x +c \right )\right )}{d \left (1+n \right )}+\frac {a \left (\sin ^{2+n}\left (d x +c \right )\right )}{d \left (2+n \right )}-\frac {2 a \left (\sin ^{3+n}\left (d x +c \right )\right )}{d \left (3+n \right )}-\frac {2 a \left (\sin ^{4+n}\left (d x +c \right )\right )}{d \left (4+n \right )}+\frac {a \left (\sin ^{5+n}\left (d x +c \right )\right )}{d \left (5+n \right )}+\frac {a \left (\sin ^{6+n}\left (d x +c \right )\right )}{d \left (6+n \right )} \]
command
integrate(cos(d*x+c)**5*sin(d*x+c)**n*(a+a*sin(d*x+c)),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 \[ \text {Timed out} \]_____________________________________________________