\[ \int \frac {-2+e^x (1+(-2-x) \log (5))+\left (-2 x+e^x x \log (5)\right ) \log (x)+\left (2+e^x (-1+(2+x) \log (5))\right ) \log (x) \log \left (\frac {e^x \log (5) \log (x)}{2+e^x (-1+(2+x) \log (5))}\right )}{\left (2 x^2+e^x \left (-x^2+\left (2 x^2+x^3\right ) \log (5)\right )\right ) \log (x)+\left (4 x+e^x \left (-2 x+\left (4 x+2 x^2\right ) \log (5)\right )\right ) \log (x) \log \left (\frac {e^x \log (5) \log (x)}{2+e^x (-1+(2+x) \log (5))}\right )+\left (2+e^x (-1+(2+x) \log (5))\right ) \log (x) \log ^2\left (\frac {e^x \log (5) \log (x)}{2+e^x (-1+(2+x) \log (5))}\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{x +\ln \! \left (\frac {\ln \left (x \right )}{x -\frac {1-2 \,{\mathrm e}^{-x}}{\ln \left (5\right )}+2}\right )}+2 \]
command
Integrate[(-2 + E^x*(1 + (-2 - x)*Log[5]) + (-2*x + E^x*x*Log[5])*Log[x] + (2 + E^x*(-1 + (2 + x)*Log[5]))*Log[x]*Log[(E^x*Log[5]*Log[x])/(2 + E^x*(-1 + (2 + x)*Log[5]))])/((2*x^2 + E^x*(-x^2 + (2*x^2 + x^3)*Log[5]))*Log[x] + (4*x + E^x*(-2*x + (4*x + 2*x^2)*Log[5]))*Log[x]*Log[(E^x*Log[5]*Log[x])/(2 + E^x*(-1 + (2 + x)*Log[5]))] + (2 + E^x*(-1 + (2 + x)*Log[5]))*Log[x]*Log[(E^x*Log[5]*Log[x])/(2 + E^x*(-1 + (2 + x)*Log[5]))]^2),x]
Mathematica 13.1 output
\[ \int \frac {-2+e^x (1+(-2-x) \log (5))+\left (-2 x+e^x x \log (5)\right ) \log (x)+\left (2+e^x (-1+(2+x) \log (5))\right ) \log (x) \log \left (\frac {e^x \log (5) \log (x)}{2+e^x (-1+(2+x) \log (5))}\right )}{\left (2 x^2+e^x \left (-x^2+\left (2 x^2+x^3\right ) \log (5)\right )\right ) \log (x)+\left (4 x+e^x \left (-2 x+\left (4 x+2 x^2\right ) \log (5)\right )\right ) \log (x) \log \left (\frac {e^x \log (5) \log (x)}{2+e^x (-1+(2+x) \log (5))}\right )+\left (2+e^x (-1+(2+x) \log (5))\right ) \log (x) \log ^2\left (\frac {e^x \log (5) \log (x)}{2+e^x (-1+(2+x) \log (5))}\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {x}{x+\log \left (\frac {e^x \log (5) \log (x)}{2+e^x (-1+(2+x) \log (5))}\right )} \]