\[ \int \frac {e^{-x^2} x \left (2 x^2+e^{2+2 e^{e^3}} \left (-2+2 x^2\right )+e^2 \left (-8+6 x^2+2 x^4\right )+e^{e^{e^3}} \left (-2 x+e^2 \left (4 x-4 x^3\right )\right )\right )+e^{-x^2} x \left (8-6 x^2-2 x^4+e^{2 e^{e^3}} \left (2-2 x^2\right )+e^{e^{e^3}} \left (-4 x+4 x^3\right )\right ) \log \left (4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2\right )}{4+e^{2 e^{e^3}}-2 e^{e^{e^3}} x+x^2} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\ln \left (x \right )-x^{2}} \left (\ln \left (\left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3}}}-x \right )^{2}+4\right )-{\mathrm e}^{2}\right ) x \]
command
integrate((((-2*x**2+2)*exp(exp(exp(3)))**2+(4*x**3-4*x)*exp(exp(exp(3)))-2*x**4-6*x**2+8)*exp(ln(x)-x**2)*ln(exp(exp(exp(3)))**2-2*x*exp(exp(exp(3)))+x**2+4)+((2*x**2-2)*exp(2)*exp(exp(exp(3)))**2+((-4*x**3+4*x)*exp(2)-2*x)*exp(exp(exp(3)))+(2*x**4+6*x**2-8)*exp(2)+2*x**2)*exp(ln(x)-x**2))/(exp(exp(exp(3)))**2-2*x*exp(exp(exp(3)))+x**2+4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \left (x^{2} \log {\left (x^{2} - 2 x e^{e^{e^{3}}} + 4 + e^{2 e^{e^{3}}} \right )} - x^{2} e^{2}\right ) e^{- x^{2}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________