\[ \int \frac {e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x-2 x \log (3)-4 e^{2-x^2} x^2 \log (3)+\left (1+e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x\right ) \log \left (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} \left (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x\right )\right )}{\left (x+e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x^2\right ) \log \left (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} \left (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x\right )\right ) \log \left (x \log \left (e^{2 e^{2-x^2} \log (3)-2 x \log (3)} \left (-1-e^{-2 e^{2-x^2} \log (3)+2 x \log (3)} x\right )\right )\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\ln \! \left (x \ln \! \left (-{\mathrm e}^{-2 \ln \left (3\right ) \left (x -{\mathrm e}^{-x^{2}+2}\right )}-x \right )\right )\right ) \]
command
integrate(((x*exp(-log(3)*exp(-x^2+2)+x*log(3))^2+1)*log((-x*exp(-log(3)*exp(-x^2+2)+x*log(3))^2-1)/exp(-log(3)*exp(-x^2+2)+x*log(3))^2)+x*exp(-log(3)*exp(-x^2+2)+x*log(3))^2-4*x^2*log(3)*exp(-x^2+2)-2*x*log(3))/(x^2*exp(-log(3)*exp(-x^2+2)+x*log(3))^2+x)/log((-x*exp(-log(3)*exp(-x^2+2)+x*log(3))^2-1)/exp(-log(3)*exp(-x^2+2)+x*log(3))^2)/log(x*log((-x*exp(-log(3)*exp(-x^2+2)+x*log(3))^2-1)/exp(-log(3)*exp(-x^2+2)+x*log(3))^2)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \log \left (\log \left (x \log \left (-{\left (x e^{\left (2 \, x \log \left (3\right ) - 2 \, e^{\left (-x^{2} + 2\right )} \log \left (3\right )\right )} + 1\right )} e^{\left (-2 \, x \log \left (3\right ) + 2 \, e^{\left (-x^{2} + 2\right )} \log \left (3\right )\right )}\right )\right )\right ) \]