\[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{x +\ln \left (x^{2}\right )^{2} \left (\frac {2}{x}+x +\ln \left (x \,{\mathrm e}^{\frac {4}{5}}\right )\right )} \]
command
integrate((4*x*log(x^2)*log(x*exp(4/5))+(x^2+x-2)*log(x^2)^2+(4*x^2+8)*log(x^2)+x^2)*exp((x*log(x^2)^2*log(x*exp(4/5))+(x^2+2)*log(x^2)^2+x^2)/x)/x^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ e^{\left (x \log \left (x^{2}\right )^{2} + \log \left (x^{2}\right )^{2} \log \left (x e^{\frac {4}{5}}\right ) + x + \frac {2 \, \log \left (x^{2}\right )^{2}}{x}\right )} \]