\[ \int \frac {2 e^{-11+x-x^2}+e^{-11+x-x^2} \left (2 x-3 x^2-2 x^3\right ) \log \left (\frac {x}{4+2 x}\right )}{e^2 \left (2 x+x^2\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\frac {x}{4+2 x}\right ) {\mathrm e}^{-2} {\mathrm e}^{-x^{2}+x -11} \]
command
Int[(2*E^(-11 + x - x^2) + E^(-11 + x - x^2)*(2*x - 3*x^2 - 2*x^3)*Log[x/(4 + 2*x)])/(E^2*(2*x + x^2)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {2 e^{-11+x-x^2}+e^{-11+x-x^2} \left (2 x-3 x^2-2 x^3\right ) \log \left (\frac {x}{4+2 x}\right )}{e^2 \left (2 x+x^2\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {e^{-x^2+x-13} \left (-2 x^2-3 x+2\right ) \log \left (\frac {x}{2 (x+2)}\right )}{(1-2 x) (x+2)} \]