\[ \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {24 b^{4} n^{4} x}{64 b^{4} n^{4}+20 b^{2} n^{2}+1}+\frac {12 b^{2} n^{2} x \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{64 b^{4} n^{4}+20 b^{2} n^{2}+1}+\frac {x \left (\cos ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{16 b^{2} n^{2}+1}+\frac {24 b^{3} n^{3} x \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{64 b^{4} n^{4}+20 b^{2} n^{2}+1}+\frac {4 b n x \left (\cos ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{16 b^{2} n^{2}+1} \]
command
integrate(cos(a+b*log(c*x^n))^4,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 \[ \text {Timed out} \]_______________________________________________________