\[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (\sin ^{1+n}\left (d x +c \right )\right )}{d \left (1+n \right )}+\frac {2 a^{2} \left (\sin ^{2+n}\left (d x +c \right )\right )}{d \left (2+n \right )}-\frac {2 a^{2} \left (\sin ^{3+n}\left (d x +c \right )\right )}{d \left (3+n \right )}-\frac {6 a^{2} \left (\sin ^{4+n}\left (d x +c \right )\right )}{d \left (4+n \right )}+\frac {6 a^{2} \left (\sin ^{6+n}\left (d x +c \right )\right )}{d \left (6+n \right )}+\frac {2 a^{2} \left (\sin ^{7+n}\left (d x +c \right )\right )}{d \left (7+n \right )}-\frac {2 a^{2} \left (\sin ^{8+n}\left (d x +c \right )\right )}{d \left (8+n \right )}-\frac {a^{2} \left (\sin ^{9+n}\left (d x +c \right )\right )}{d \left (9+n \right )} \]
command
integrate(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________