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