\[ \int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx \]
Optimal antiderivative \[ -\frac {\left (a^{2}-b^{2}\right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{1+m}}{b^{7} d \left (1+m \right )}+\frac {6 a \left (a^{2}-b^{2}\right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{2+m}}{b^{7} d \left (2+m \right )}-\frac {3 \left (5 a^{4}-6 a^{2} b^{2}+b^{4}\right ) \left (a +b \sin \left (d x +c \right )\right )^{3+m}}{b^{7} d \left (3+m \right )}+\frac {4 a \left (5 a^{2}-3 b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{4+m}}{b^{7} d \left (4+m \right )}-\frac {3 \left (5 a^{2}-b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{5+m}}{b^{7} d \left (5+m \right )}+\frac {6 a \left (a +b \sin \left (d x +c \right )\right )^{6+m}}{b^{7} d \left (6+m \right )}-\frac {\left (a +b \sin \left (d x +c \right )\right )^{7+m}}{b^{7} d \left (7+m \right )} \]
command
integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,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 \[ \text {Timed out} \]_______________________________________________________