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