\[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx \]
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)*ln(x)+(2-2*x)*exp(x)-2*x**3+2*x**2)*ln(ln(x)+x**2+exp(x))*ln(ln(ln(x)+x**2+exp(x)))+(4*exp(x)*x+8*x**2+4)*ln(x)-4*exp(x)*x**2-8*x**3-4*x)/(x*ln(x)**2+(exp(x)*x+x**3-x**2)*ln(x)-exp(x)*x**2-x**4)/ln(ln(x)+x**2+exp(x))/ln(ln(ln(x)+x**2+exp(x))),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ 2 \log {\left (- x + \log {\left (x \right )} \right )} + 4 \log {\left (\log {\left (\log {\left (x^{2} + e^{x} + \log {\left (x \right )} \right )} \right )} \right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________