\[ \int \frac {e^x (-4+2 x) \log (-2+x)+\left (-e^x x+e^x (-2+x) \log (-2+x)\right ) \log \left (x^2\right )+e^x \left (2 x^2-x^3\right ) \log ^2\left (x^2\right )+\left (e^x (-2+x) \log (-2+x) \log \left (x^2\right )+e^x \left (-2 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )+\left (e^x \left (2-3 x+x^2\right ) \log (-2+x) \log \left (x^2\right )+e^x \left (2 x^2-3 x^3+x^4\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right ) \log \left (\frac {x}{\log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}\right )}{\left (\left (-2 x^2+x^3\right ) \log (-2+x) \log \left (x^2\right )+\left (-2 x^4+x^5\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {\log (-2+x)+x^2 \log \left (x^2\right )}{x \log \left (x^2\right )}\right )} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{x} \ln \! \left (\frac {x}{\ln \left (\frac {\ln \left (-2+x \right )}{x \ln \left (x^{2}\right )}+x \right )}\right )}{x} \]
command
Int[(E^x*(-4 + 2*x)*Log[-2 + x] + (-(E^x*x) + E^x*(-2 + x)*Log[-2 + x])*Log[x^2] + E^x*(2*x^2 - x^3)*Log[x^2]^2 + (E^x*(-2 + x)*Log[-2 + x]*Log[x^2] + E^x*(-2*x^2 + x^3)*Log[x^2]^2)*Log[(Log[-2 + x] + x^2*Log[x^2])/(x*Log[x^2])] + (E^x*(2 - 3*x + x^2)*Log[-2 + x]*Log[x^2] + E^x*(2*x^2 - 3*x^3 + x^4)*Log[x^2]^2)*Log[(Log[-2 + x] + x^2*Log[x^2])/(x*Log[x^2])]*Log[x/Log[(Log[-2 + x] + x^2*Log[x^2])/(x*Log[x^2])]])/(((-2*x^2 + x^3)*Log[-2 + x]*Log[x^2] + (-2*x^4 + x^5)*Log[x^2]^2)*Log[(Log[-2 + x] + x^2*Log[x^2])/(x*Log[x^2])]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {e^x \log \left (\frac {x}{\log \left (\frac {\log (x-2)}{x \log \left (x^2\right )}+x\right )}\right )}{x} \]