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