\[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx \]
Optimal antiderivative \[ 1+{\left (3-\frac {\ln \! \left (\ln \! \left (\left (10+2 \,{\mathrm e}^{x}-2 x \right ) x +\ln \! \left (x \right )\right )\right )}{\ln \! \left (x \right )}\right )}^{2} \]
command
Int[((-6 - 60*x + 24*x^2 + E^x*(-12*x - 12*x^2))*Log[x]^2 + ((2 + 20*x - 8*x^2 + E^x*(4*x + 4*x^2))*Log[x] + ((60*x + 12*E^x*x - 12*x^2)*Log[x] + 6*Log[x]^2)*Log[10*x + 2*E^x*x - 2*x^2 + Log[x]])*Log[Log[10*x + 2*E^x*x - 2*x^2 + Log[x]]] + (-20*x - 4*E^x*x + 4*x^2 - 2*Log[x])*Log[10*x + 2*E^x*x - 2*x^2 + Log[x]]*Log[Log[10*x + 2*E^x*x - 2*x^2 + Log[x]]]^2)/(((10*x^2 + 2*E^x*x^2 - 2*x^3)*Log[x]^3 + x*Log[x]^4)*Log[10*x + 2*E^x*x - 2*x^2 + Log[x]]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\left (-6-60 x+24 x^2+e^x \left (-12 x-12 x^2\right )\right ) \log ^2(x)+\left (\left (2+20 x-8 x^2+e^x \left (4 x+4 x^2\right )\right ) \log (x)+\left (\left (60 x+12 e^x x-12 x^2\right ) \log (x)+6 \log ^2(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right ) \log \left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )+\left (-20 x-4 e^x x+4 x^2-2 \log (x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right ) \log ^2\left (\log \left (10 x+2 e^x x-2 x^2+\log (x)\right )\right )}{\left (\left (10 x^2+2 e^x x^2-2 x^3\right ) \log ^3(x)+x \log ^4(x)\right ) \log \left (10 x+2 e^x x-2 x^2+\log (x)\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {\log ^2\left (\log \left (2 \left (-x+e^x+5\right ) x+\log (x)\right )\right )}{\log ^2(x)}-\frac {6 \log \left (\log \left (2 \left (-x+e^x+5\right ) x+\log (x)\right )\right )}{\log (x)} \]