\[ \int \frac {e^{\frac {-e+\log (3)}{-20-x+x \log (x)}} (e-\log (3)) \log (x)}{400+40 x+x^2+\left (-40 x-2 x^2\right ) \log (x)+x^2 \log ^2(x)} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {-\ln \left (3\right )+{\mathrm e}}{20+x -x \ln \left (x \right )}} \]
command
Int[(E^((-E + Log[3])/(-20 - x + x*Log[x]))*(E - Log[3])*Log[x])/(400 + 40*x + x^2 + (-40*x - 2*x^2)*Log[x] + x^2*Log[x]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ 3^{\frac {1}{-x+x \log (x)-20}} e^{\frac {e}{x+x (-\log (x))+20}} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {e^{\frac {-e+\log (3)}{-20-x+x \log (x)}} (e-\log (3)) \log (x)}{400+40 x+x^2+\left (-40 x-2 x^2\right ) \log (x)+x^2 \log ^2(x)} \, dx \]________________________________________________________________________________________