\[ \int \frac {e^{-e^x} \left (\left (20+x^2\right ) \log (2)+e^x \left (-20 x+x^2+x^3\right ) \log (2)\right )}{2000-200 x-195 x^2+10 x^3+5 x^4} \, dx \]
Optimal antiderivative \[ \frac {\ln \! \left (2\right ) {\mathrm e}^{-{\mathrm e}^{x}} x}{5 \left (5+x \right ) \left (-x +4\right )} \]
command
Int[((20 + x^2)*Log[2] + E^x*(-20*x + x^2 + x^3)*Log[2])/(E^E^x*(2000 - 200*x - 195*x^2 + 10*x^3 + 5*x^4)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{-e^x} \left (\left (20+x^2\right ) \log (2)+e^x \left (-20 x+x^2+x^3\right ) \log (2)\right )}{2000-200 x-195 x^2+10 x^3+5 x^4} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {e^{-e^x} \left (-x^3-x^2+20 x\right ) \log (2)}{5 \left (x^4+2 x^3-39 x^2-40 x+400\right )} \]