\[ \int \frac {2 e^{-4+x+\frac {2 \left (2+5 x^2\right )}{x}} x^2 \log \left (1+e^{-4+x}\right )+e^{\frac {2 \left (2+5 x^2\right )}{x}} \left (-4+10 x^2+e^{-4+x} \left (-4+10 x^2\right )\right ) \log ^2\left (1+e^{-4+x}\right )}{e^{\frac {2 \left (2+5 x^2\right )}{x}} \left (x^2+e^{-4+x} x^2\right ) \log ^2\left (1+e^{-4+x}\right )+\left (-2 x^2-2 e^{-4+x} x^2\right ) \log (\log (4))} \, dx \]
Optimal antiderivative \[ \ln \left (\frac {\ln \left ({\mathrm e}^{x -4}+1\right )^{2} {\mathrm e}^{\frac {4}{x}+10 x}}{2}-\ln \left (2 \ln \left (2\right )\right )\right ) \]
command
Int[(2*E^(-4 + x + (2*(2 + 5*x^2))/x)*x^2*Log[1 + E^(-4 + x)] + E^((2*(2 + 5*x^2))/x)*(-4 + 10*x^2 + E^(-4 + x)*(-4 + 10*x^2))*Log[1 + E^(-4 + x)]^2)/(E^((2*(2 + 5*x^2))/x)*(x^2 + E^(-4 + x)*x^2)*Log[1 + E^(-4 + x)]^2 + (-2*x^2 - 2*E^(-4 + x)*x^2)*Log[Log[4]]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \log \left (e^{10 x+\frac {4}{x}} \log ^2\left (e^{x-4}+1\right )-2 \log (\log (4))\right ) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]________________________________________________________________________________________