For a bit of context this integral came up during my studies for multivariate analysis. It is a seemingly simple looking integral that has blocked all attempts to pry into its secrets.
$\int \ln(x)^{\exp(x)} dx$
I've attempted to use tricks like $u$-substitutions and integration by parts. I've even derived a formula for $\frac{d f(x)^{g(x)}}{dx}$ and attempted to use that with no good results. The u-subs themselves were $u = e^x$ and $u = \ln(x)$ and they did not lead in good directions. Giving other janky integrals that were difficult to solve. Wolfram alpha did not give me any hints either and at this point I am considering using the tools I've learned in multivariable analysis, taking a page out of the $\int \exp(-x^2) dx$ playbook and mashing some coordinate system onto it to bruteforce a solution. Please help me.