\[ \int x \sin ^4\left (a+b \log \left (c x^n\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {3 b^{4} n^{4} x^{2}}{4 \left (4 b^{4} n^{4}+5 b^{2} n^{2}+1\right )}-\frac {3 b^{3} n^{3} x^{2} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 \left (4 b^{4} n^{4}+5 b^{2} n^{2}+1\right )}+\frac {3 b^{2} n^{2} x^{2} \left (\sin ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2 \left (4 b^{4} n^{4}+5 b^{2} n^{2}+1\right )}-\frac {b n \,x^{2} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \left (\sin ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{4 b^{2} n^{2}+1}+\frac {x^{2} \left (\sin ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{8 b^{2} n^{2}+2} \]
command
integrate(x*sin(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 \[ \text {Timed out} \]_______________________________________________________