\[ \int x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {12 b^{2} n^{2} x^{2} \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{9 b^{4} n^{4}+40 b^{2} n^{2}+16}+\frac {2 x^{2} \left (\cos ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{9 b^{2} n^{2}+4}+\frac {6 b^{3} n^{3} x^{2} \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{9 b^{4} n^{4}+40 b^{2} n^{2}+16}+\frac {3 b n \,x^{2} \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{9 b^{2} n^{2}+4} \]
command
integrate(x*cos(a+b*log(c*x^n))^3,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 \[ \text {Timed out} \]_______________________________________________________