\[ \int \frac {e^{e^x} \left (e^x \left (8+2 e^{x-x^2}\right )+e^{x-x^2} (-1+2 x)\right )+e^{e^x+x} \left (-4-e^{x-x^2}\right ) \log \left (4+e^{x-x^2}\right )}{4+e^{x-x^2}} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{{\mathrm e}^{x}} \left (2-\ln \! \left ({\mathrm e}^{-x^{2}+x}+4\right )\right )+\ln \! \left (\frac {2}{5}\right ) \]
command
Int[(E^E^x*(E^x*(8 + 2*E^(x - x^2)) + E^(x - x^2)*(-1 + 2*x)) + E^(E^x + x)*(-4 - E^(x - x^2))*Log[4 + E^(x - x^2)])/(4 + E^(x - x^2)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{e^x} \left (e^x \left (8+2 e^{x-x^2}\right )+e^{x-x^2} (-1+2 x)\right )+e^{e^x+x} \left (-4-e^{x-x^2}\right ) \log \left (4+e^{x-x^2}\right )}{4+e^{x-x^2}} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ 2 e^{e^x}-e^{e^x} \log \left (e^{x-x^2}+4\right ) \]