\[ \int \frac {e^{-4+\frac {-2 x+e^4 \log \left (x+x^2\right ) \log (\log (x))}{e^4 \log \left (x+x^2\right )}} \left (\left (12 x+24 x^2\right ) \log (x)+\left (-12 x-12 x^2\right ) \log (x) \log \left (x+x^2\right )+e^4 (6+6 x) \log ^2\left (x+x^2\right )\right )}{\left (x+x^2\right ) \log (x) \log ^2\left (x+x^2\right )} \, dx \]
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
integrate(((6+6*x)*exp(4)*log(x^2+x)^2+(-12*x^2-12*x)*log(x)*log(x^2+x)+(24*x^2+12*x)*log(x))*exp((exp(4)*log(x^2+x)*log(log(x))-2*x)/exp(4)/log(x^2+x))/(x^2+x)/exp(4)/log(x)/log(x^2+x)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ 6 \, e^{\left (-\frac {2 \, x e^{\left (-4\right )}}{\log \left (x^{2} + x\right )} + \log \left (\log \left (x\right )\right )\right )} \]