\[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx \]
Optimal antiderivative \[ \frac {2 x}{x^{2} \left ({\mathrm e}^{5}-x^{2} {\mathrm e}^{8}\right )-\ln \left (\left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )} \]
command
integrate(((-2*log(x)*log(log(x))+2*x*log(x))*log(log(log(x))^2-2*x*log(log(x))+x^2)+(-2*x^2*exp(5)+6*x^4*exp(4)^2)*log(x)*log(log(x))+(2*x^3*exp(5)-6*x^5*exp(4)^2-4*x)*log(x)+4)/((log(x)*log(log(x))-x*log(x))*log(log(log(x))^2-2*x*log(log(x))+x^2)^2+((-2*x^2*exp(5)+2*x^4*exp(4)^2)*log(x)*log(log(x))+(2*x^3*exp(5)-2*x^5*exp(4)^2)*log(x))*log(log(log(x))^2-2*x*log(log(x))+x^2)+(x^4*exp(5)^2-2*x^6*exp(4)^2*exp(5)+x^8*exp(4)^4)*log(x)*log(log(x))+(-x^5*exp(5)^2+2*x^7*exp(4)^2*exp(5)-x^9*exp(4)^4)*log(x)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, x}{x^{4} e^{8} - x^{2} e^{5} + \log \left (x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________