\[ \int \frac {e^{\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}} \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {4 \,{\mathrm e}^{-\frac {2}{3}}}{3}+\frac {\ln \left (\ln \left (x^{2}\right )\right )-2}{x}-2 x} \]
command
Int[(E^((4*x + E^(2/3)*(-6 - 6*x^2) + 3*E^(2/3)*Log[Log[x^2]])/(3*E^(2/3)*x))*(2 + (2 - 2*x^2)*Log[x^2] - Log[x^2]*Log[Log[x^2]]))/(x^2*Log[x^2]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \exp \left (\frac {2 \left (2 x-3 e^{2/3} \left (x^2+1\right )\right )}{3 e^{2/3} x}\right ) \sqrt [x]{\log \left (x^2\right )} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {\exp \left (\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}\right ) \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\right )} \, dx \]________________________________________________________________________________________