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