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