\[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx \]
Optimal antiderivative \[ \frac {8 \left (a +a \sin \left (d x +c \right )\right )^{4+m}}{a^{4} d \left (4+m \right )}-\frac {12 \left (a +a \sin \left (d x +c \right )\right )^{5+m}}{a^{5} d \left (5+m \right )}+\frac {6 \left (a +a \sin \left (d x +c \right )\right )^{6+m}}{a^{6} d \left (6+m \right )}-\frac {\left (a +a \sin \left (d x +c \right )\right )^{7+m}}{a^{7} d \left (7+m \right )} \]
command
integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^m,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{3} - 6 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{3} + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{3} - 8 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{3} + 15 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} - 96 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{2} + 204 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{2} - 144 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m^{2} + 74 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m - 498 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m + 1128 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m - 856 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3} m + 120 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{7} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} - 840 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{6} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a + 2016 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} - 1680 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{3}}{{\left (a^{6} m^{4} + 22 \, a^{6} m^{3} + 179 \, a^{6} m^{2} + 638 \, a^{6} m + 840 \, a^{6}\right )} a d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________