\[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx \]
Optimal antiderivative \[ {\left (2-\ln \! \left (\frac {x}{\ln \! \left (\frac {5 \ln \left (3\right ) {\mathrm e}^{-2}}{x}\right )}-x^{2}-x \right )\right )}^{2} \]
command
Int[(4 + 4*Log[(5*Log[3])/(E^2*x)] + (-4 - 8*x)*Log[(5*Log[3])/(E^2*x)]^2 + (-2 - 2*Log[(5*Log[3])/(E^2*x)] + (2 + 4*x)*Log[(5*Log[3])/(E^2*x)]^2)*Log[(x + (-x - x^2)*Log[(5*Log[3])/(E^2*x)])/Log[(5*Log[3])/(E^2*x)]])/(-(x*Log[(5*Log[3])/(E^2*x)]) + (x + x^2)*Log[(5*Log[3])/(E^2*x)]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {4+4 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(-4-8 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )+\left (-2-2 \log \left (\frac {5 \log (3)}{e^2 x}\right )+(2+4 x) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )\right ) \log \left (\frac {x+\left (-x-x^2\right ) \log \left (\frac {5 \log (3)}{e^2 x}\right )}{\log \left (\frac {5 \log (3)}{e^2 x}\right )}\right )}{-x \log \left (\frac {5 \log (3)}{e^2 x}\right )+\left (x+x^2\right ) \log ^2\left (\frac {5 \log (3)}{e^2 x}\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \left (2-\log \left (-x \left (x+\frac {1}{2-\log \left (\frac {\log (243)}{x}\right )}+1\right )\right )\right )^2 \]