\[ \int \frac {-1+6 x-x^2+3 \log (x)}{\left (12 x^2-4 x^3+4 x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )+\left (3 x^2-x^3+x \log (x)\right ) \log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right ) \log \left (\log \left (\frac {2 x^3}{3 x-x^2+\log (x)}\right )\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (-12-3 \ln \! \left (\ln \! \left (\frac {2 x^{2}}{3+\frac {\ln \left (x \right )}{x}-x}\right )\right )\right ) \]
command
integrate((3*log(x)-x^2+6*x-1)/((x*log(x)-x^3+3*x^2)*log(2*x^3/(log(x)-x^2+3*x))*log(log(2*x^3/(log(x)-x^2+3*x)))+(4*x*log(x)-4*x^3+12*x^2)*log(2*x^3/(log(x)-x^2+3*x))),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \log \left (\log \left (i \, \pi + \log \left (2\right ) - \log \left (x^{2} - 3 \, x - \log \left (x\right )\right ) + 3 \, \log \left (x\right )\right ) + 4\right ) \]