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