\[ \int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {a \left (\sin ^{1+n}\left (d x +c \right )\right )}{d \left (1+n \right )}+\frac {a \left (\sin ^{2+n}\left (d x +c \right )\right )}{d \left (2+n \right )}-\frac {3 a \left (\sin ^{3+n}\left (d x +c \right )\right )}{d \left (3+n \right )}-\frac {3 a \left (\sin ^{4+n}\left (d x +c \right )\right )}{d \left (4+n \right )}+\frac {3 a \left (\sin ^{5+n}\left (d x +c \right )\right )}{d \left (5+n \right )}+\frac {3 a \left (\sin ^{6+n}\left (d x +c \right )\right )}{d \left (6+n \right )}-\frac {a \left (\sin ^{7+n}\left (d x +c \right )\right )}{d \left (7+n \right )}-\frac {a \left (\sin ^{8+n}\left (d x +c \right )\right )}{d \left (8+n \right )} \]
command
integrate(cos(d*x+c)^7*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^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} + 12 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} - 3 \, n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 44 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} - 42 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 48 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{8} + 3 \, n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 168 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 48 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 192 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 228 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 18 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 288 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} - 104 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} - 192 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}\right )} a}{n^{4} + 20 \, n^{3} + 140 \, n^{2} + 400 \, n + 384} + \frac {{\left (n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} + 9 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} - 3 \, n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 23 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} - 33 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 15 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{7} + 3 \, n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 93 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 39 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 63 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 141 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 15 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 105 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} - 71 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) - 105 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )\right )} a}{n^{4} + 16 \, n^{3} + 86 \, n^{2} + 176 \, n + 105}}{d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________