\[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\left (\ln \! \left (x \right )-x \right )^{2} \ln \! \left (\ln \! \left (\ln \! \left (x \right )+x^{2}+{\mathrm e}^{x}\right )\right )^{4}\right ) \]
command
integrate((((2-2*x)*log(x)+(2-2*x)*exp(x)-2*x^3+2*x^2)*log(log(x)+x^2+exp(x))*log(log(log(x)+x^2+exp(x)))+(4*exp(x)*x+8*x^2+4)*log(x)-4*exp(x)*x^2-8*x^3-4*x)/(x*log(x)^2+(exp(x)*x+x^3-x^2)*log(x)-exp(x)*x^2-x^4)/log(log(x)+x^2+exp(x))/log(log(log(x)+x^2+exp(x))),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
\[ 2 \, \log \left (-x + \log \left (x\right )\right ) + 4 \, \log \left (\log \left (\log \left (x^{2} + e^{x} + \log \left (x\right )\right )\right )\right ) \]