\[ \int \cos ^5(a+b x) (c \sin (a+b x))^m \, dx \]
Optimal antiderivative \[ \frac {\left (c \sin \left (b x +a \right )\right )^{1+m}}{b c \left (1+m \right )}-\frac {2 \left (c \sin \left (b x +a \right )\right )^{3+m}}{b \,c^{3} \left (3+m \right )}+\frac {\left (c \sin \left (b x +a \right )\right )^{5+m}}{b \,c^{5} \left (5+m \right )} \]
command
integrate(cos(b*x+a)^5*(c*sin(b*x+a))^m,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\left (c \sin \left (b x + a\right )\right )^{m} c^{5} m^{2} \sin \left (b x + a\right )^{5} + 4 \, \left (c \sin \left (b x + a\right )\right )^{m} c^{5} m \sin \left (b x + a\right )^{5} - 2 \, \left (c \sin \left (b x + a\right )\right )^{m} c^{5} m^{2} \sin \left (b x + a\right )^{3} + 3 \, \left (c \sin \left (b x + a\right )\right )^{m} c^{5} \sin \left (b x + a\right )^{5} - 12 \, \left (c \sin \left (b x + a\right )\right )^{m} c^{5} m \sin \left (b x + a\right )^{3} + \left (c \sin \left (b x + a\right )\right )^{m} c^{5} m^{2} \sin \left (b x + a\right ) - 10 \, \left (c \sin \left (b x + a\right )\right )^{m} c^{5} \sin \left (b x + a\right )^{3} + 8 \, \left (c \sin \left (b x + a\right )\right )^{m} c^{5} m \sin \left (b x + a\right ) + 15 \, \left (c \sin \left (b x + a\right )\right )^{m} c^{5} \sin \left (b x + a\right )}{{\left (c^{4} m^{3} + 9 \, c^{4} m^{2} + 23 \, c^{4} m + 15 \, c^{4}\right )} b c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________