\[ \int \frac {3^{-1/x} \sqrt [x]{\log \left (x^4\right )} \left (-48+12 \log \left (x^4\right ) \log \left (\frac {\log \left (x^4\right )}{3}\right )\right )}{\left (625 x^2+50 x^2 \log (4)+x^2 \log ^2(4)\right ) \log \left (x^4\right )+3^{-1/x} \left (-50 x^2-2 x^2 \log (4)\right ) \log ^{1+\frac {1}{x}}\left (x^4\right )+3^{-2/x} x^2 \log ^{1+\frac {2}{x}}\left (x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {12}{-25-2 \ln \! \left (2\right )+{\mathrm e}^{\frac {\ln \left (\frac {\ln \left (x^{4}\right )}{3}\right )}{x}}} \]
command
Integrate[(Log[x^4]^x^(-1)*(-48 + 12*Log[x^4]*Log[Log[x^4]/3]))/(3^x^(-1)*((625*x^2 + 50*x^2*Log[4] + x^2*Log[4]^2)*Log[x^4] + ((-50*x^2 - 2*x^2*Log[4])*Log[x^4]^(1 + x^(-1)))/3^x^(-1) + (x^2*Log[x^4]^(1 + 2/x))/3^(2/x))),x]
Mathematica 13.1 output
\[ \int \frac {3^{-1/x} \sqrt [x]{\log \left (x^4\right )} \left (-48+12 \log \left (x^4\right ) \log \left (\frac {\log \left (x^4\right )}{3}\right )\right )}{\left (625 x^2+50 x^2 \log (4)+x^2 \log ^2(4)\right ) \log \left (x^4\right )+3^{-1/x} \left (-50 x^2-2 x^2 \log (4)\right ) \log ^{1+\frac {1}{x}}\left (x^4\right )+3^{-2/x} x^2 \log ^{1+\frac {2}{x}}\left (x^4\right )} \, dx \]
Mathematica 12.3 output
\[ -\frac {4\ 3^{1+\frac {1}{x}}}{3^{\frac {1}{x}} (25+\log (4))-\sqrt [x]{\log \left (x^4\right )}} \]