\[ \int \frac {\left (16-12 x^2+4 x^3+e^3 \left (-16 x^2-16 x^3+28 x^4-8 x^5\right )+e^3 \left (-16 x-16 x^2+28 x^3-8 x^4\right ) \log (x)\right ) \log \left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )+\left (-4 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )} \, dx \]
Optimal antiderivative \[ \left (-2+x \right )^{2} \ln \left (\ln \left (\left (x +\ln \left (x \right )\right ) {\mathrm e}^{-\left (x^{2}+x +2\right ) {\mathrm e}^{3}}\right )^{2}\right )^{2} \]
command
integrate((((2*x^2-4*x)*log(x)+2*x^3-4*x^2)*log((x+log(x))/exp((x^2+x+2)*exp(3)))*log(log((x+log(x))/exp((x^2+x+2)*exp(3)))^2)^2+((-8*x^4+28*x^3-16*x^2-16*x)*exp(3)*log(x)+(-8*x^5+28*x^4-16*x^3-16*x^2)*exp(3)+4*x^3-12*x^2+16)*log(log((x+log(x))/exp((x^2+x+2)*exp(3)))^2))/(x*log(x)+x^2)/log((x+log(x))/exp((x^2+x+2)*exp(3))),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ x^{2} \log \left (x^{4} e^{6} + 2 \, x^{3} e^{6} - 2 \, x^{2} e^{3} \log \left (x + \log \left (x\right )\right ) + 5 \, x^{2} e^{6} - 2 \, x e^{3} \log \left (x + \log \left (x\right )\right ) + 4 \, x e^{6} - 4 \, e^{3} \log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right )\right )^{2} + 4 \, e^{6}\right )^{2} - 4 \, x \log \left (x^{4} e^{6} + 2 \, x^{3} e^{6} - 2 \, x^{2} e^{3} \log \left (x + \log \left (x\right )\right ) + 5 \, x^{2} e^{6} - 2 \, x e^{3} \log \left (x + \log \left (x\right )\right ) + 4 \, x e^{6} - 4 \, e^{3} \log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right )\right )^{2} + 4 \, e^{6}\right )^{2} + 4 \, \log \left (x^{4} e^{6} + 2 \, x^{3} e^{6} - 2 \, x^{2} e^{3} \log \left (x + \log \left (x\right )\right ) + 5 \, x^{2} e^{6} - 2 \, x e^{3} \log \left (x + \log \left (x\right )\right ) + 4 \, x e^{6} - 4 \, e^{3} \log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right )\right )^{2} + 4 \, e^{6}\right )^{2} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________