\[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^m \, dx \]
Optimal antiderivative \[ \frac {4 \left (a +a \sin \left (d x +c \right )\right )^{3+m}}{a^{3} d \left (3+m \right )}-\frac {4 \left (a +a \sin \left (d x +c \right )\right )^{4+m}}{a^{4} d \left (4+m \right )}+\frac {\left (a +a \sin \left (d x +c \right )\right )^{5+m}}{a^{5} d \left (5+m \right )} \]
command
integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^m,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m^{2} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m^{2} + 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m^{2} + 7 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} m - 32 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a m + 36 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2} m + 12 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} - 60 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a + 80 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} a^{2}}{{\left (a^{4} m^{3} + 12 \, a^{4} m^{2} + 47 \, a^{4} m + 60 \, a^{4}\right )} a d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________