\[ \int x^m \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {2 b^{2} n^{2} x^{1+m}}{\left (1+m \right ) \left (\left (1+m \right )^{2}+4 b^{2} n^{2}\right )}+\frac {\left (1+m \right ) x^{1+m} \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{\left (1+m \right )^{2}+4 b^{2} n^{2}}+\frac {2 b n \,x^{1+m} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (1+m \right )^{2}+4 b^{2} n^{2}} \]
command
integrate(x^m*cos(a+b*log(c*x^n))^2,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 \[ \text {Timed out} \]_______________________________________________________