\[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx \]
Optimal antiderivative \[ -82-\frac {16 \,{\mathrm e}^{{\mathrm e}^{x}}}{\ln \left (\frac {\ln \left (\ln \left (\ln \left (x \right )\right )\right )}{x}\right )} \]
command
Int[(16*E^E^x - 16*E^E^x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]] - 16*E^(E^x + x)*x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]]*Log[Log[Log[Log[x]]]/x])/(x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]]*Log[Log[Log[Log[x]]]/x]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ -\frac {16 e^{e^x}}{\log \left (\frac {\log (\log (\log (x)))}{x}\right )} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx \]________________________________________________________________________________________