\[ \int \frac {e^{-\frac {2 \left (-5 e^{e^{x^2}}-5 x+x^2\right )}{e^{e^{x^2}}+x}} \left (2 e^{2 e^{x^2}} \log (x+\log (2))+2 x^2 \log (x+\log (2))+\left (-2 x^3-2 x^2 \log (2)\right ) \log ^2(x+\log (2))+e^{e^{x^2}} \left (4 x \log (x+\log (2))+\left (-4 x^2-4 x \log (2)+e^{x^2} \left (4 x^4+4 x^3 \log (2)\right )\right ) \log ^2(x+\log (2))\right )\right )}{x^3+x^2 \log (2)+e^{2 e^{x^2}} (x+\log (2))+e^{e^{x^2}} \left (2 x^2+2 x \log (2)\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\ln \! \left (2\right )+x \right )^{2} {\mathrm e}^{-\frac {2 x^{2}}{{\mathrm e}^{{\mathrm e}^{x^{2}}}+x}+10}-5 \]
command
Int[(2*E^(2*E^x^2)*Log[x + Log[2]] + 2*x^2*Log[x + Log[2]] + (-2*x^3 - 2*x^2*Log[2])*Log[x + Log[2]]^2 + E^E^x^2*(4*x*Log[x + Log[2]] + (-4*x^2 - 4*x*Log[2] + E^x^2*(4*x^4 + 4*x^3*Log[2]))*Log[x + Log[2]]^2))/(E^((2*(-5*E^E^x^2 - 5*x + x^2))/(E^E^x^2 + x))*(x^3 + x^2*Log[2] + E^(2*E^x^2)*(x + Log[2]) + E^E^x^2*(2*x^2 + 2*x*Log[2]))),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{-\frac {2 \left (-5 e^{e^{x^2}}-5 x+x^2\right )}{e^{e^{x^2}}+x}} \left (2 e^{2 e^{x^2}} \log (x+\log (2))+2 x^2 \log (x+\log (2))+\left (-2 x^3-2 x^2 \log (2)\right ) \log ^2(x+\log (2))+e^{e^{x^2}} \left (4 x \log (x+\log (2))+\left (-4 x^2-4 x \log (2)+e^{x^2} \left (4 x^4+4 x^3 \log (2)\right )\right ) \log ^2(x+\log (2))\right )\right )}{x^3+x^2 \log (2)+e^{2 e^{x^2}} (x+\log (2))+e^{e^{x^2}} \left (2 x^2+2 x \log (2)\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {e^{\frac {2 \left (5 e^{e^{x^2}}+(5-x) x\right )}{e^{e^{x^2}}+x}} x \left (-e^{x^2+e^{x^2}} x^2 (2 x+\log (4))+e^{e^{x^2}} (2 x+\log (4))+x (x+\log (2))\right ) \log ^2(x+\log (2))}{\left (e^{e^{x^2}}+x\right )^2 \left (\frac {10 e^{x^2+e^{x^2}} x-2 x+5}{e^{e^{x^2}}+x}-\frac {\left (2 e^{x^2+e^{x^2}} x+1\right ) \left (5 e^{e^{x^2}}+(5-x) x\right )}{\left (e^{e^{x^2}}+x\right )^2}\right ) (x+\log (2))} \]