\[ \int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx \]
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, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {{\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 6 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 8 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - 16 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} - 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 10 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}\right )} a}{n^{3} + 12 \, n^{2} + 44 \, n + 48} + \frac {{\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 4 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - 12 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) - 10 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 8 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 15 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )\right )} a}{n^{3} + 9 \, n^{2} + 23 \, n + 15}}{d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________