\[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {3}{\ln \! \left (\ln \! \left (\ln \! \left ({\mathrm e}^{-2 x^{2}+2 x}\right ) x +\frac {8 \ln \left (2\right )}{\ln \left (x \right )}\right )\right ) x} \]
command
Integrate[(-6*Log[4] + (6*x^2 - 9*x^3)*Log[x]^2 + (6*Log[4]*Log[x] + (3*x^2 - 3*x^3)*Log[x]^2)*Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log[Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]])/((-2*x^2*Log[4]*Log[x] + (-x^4 + x^5)*Log[x]^2)*Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]*Log[Log[(4*Log[4] + (2*x^2 - 2*x^3)*Log[x])/Log[x]]]^2),x]
Mathematica 13.1 output
\[ \int \frac {-6 \log (4)+\left (6 x^2-9 x^3\right ) \log ^2(x)+\left (6 \log (4) \log (x)+\left (3 x^2-3 x^3\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log \left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )}{\left (-2 x^2 \log (4) \log (x)+\left (-x^4+x^5\right ) \log ^2(x)\right ) \log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right ) \log ^2\left (\log \left (\frac {4 \log (4)+\left (2 x^2-2 x^3\right ) \log (x)}{\log (x)}\right )\right )} \, dx \]
Mathematica 12.3 output
\[ -\frac {3 \left (\log (256)-2 (-1+x) x^2 \log (x)\right ) \left (\log (16)+x^2 (-2+3 x) \log ^2(x)\right )}{x \left (-2 \log (4)+(-1+x) x^2 \log (x)\right ) \left (\log (256)+2 x^2 (-2+3 x) \log ^2(x)\right ) \log \left (\log \left (-2 (-1+x) x^2+\frac {\log (256)}{\log (x)}\right )\right )} \]