\[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx \]
Optimal antiderivative \[ \frac {\ln \! \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+x}}}+x \right )^{2} x^{2}}{27} \]
command
Int[((-2*x^2 + E^(E^E^(x + x^2) + E^(x + x^2) + x + x^2)*(2*x^2 + 4*x^3))*Log[-E^E^E^(x + x^2) + x] + (2*E^E^E^(x + x^2)*x - 2*x^2)*Log[-E^E^E^(x + x^2) + x]^2)/(27*E^E^E^(x + x^2) - 27*x),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {1}{27} x^2 \log ^2\left (x-e^{e^{e^{x^2+x}}}\right ) \]