\[ \int \frac {-2 x-2 x^2+e^x (2+2 x)+e^x (-2-2 x) \log \left (\frac {1+x}{5}\right )+\log \left (\frac {x^2}{3}\right ) \left (-2 x-2 x^2+e^x \left (1+x+x^2\right )+e^x \left (-1-2 x-x^2\right ) \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{(1+x) \log \left (\frac {x^2}{3}\right )} \, dx \]
Optimal antiderivative \[ \left ({\mathrm e}^{x}-x -{\mathrm e}^{x} \ln \left (\frac {x}{5}+\frac {1}{5}\right )\right ) \ln \left (\ln \left (\frac {x^{2}}{3}\right )\right ) x \]
command
integrate((((-x**2-2*x-1)*exp(x)*ln(1/5*x+1/5)+(x**2+x+1)*exp(x)-2*x**2-2*x)*ln(1/3*x**2)*ln(ln(1/3*x**2))+(-2-2*x)*exp(x)*ln(1/5*x+1/5)+(2+2*x)*exp(x)-2*x**2-2*x)/(1+x)/ln(1/3*x**2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - x^{2} \log {\left (\log {\left (\frac {x^{2}}{3} \right )} \right )} + \left (- x \log {\left (\frac {x}{5} + \frac {1}{5} \right )} \log {\left (\log {\left (\frac {x^{2}}{3} \right )} \right )} + x \log {\left (\log {\left (\frac {x^{2}}{3} \right )} \right )}\right ) e^{x} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________