Re-writing integral of x^x

Hello and thanks to those reading my question.

I don't know if this can be done, but I'm trying to re-write the integral of x^x in the form of an integral of R(x) e^g(x) where R(x) and g(x) are rational functions.

Starting from x^x = e^(x ln(x)) I have tried various substitutions (eg. u = ln(x), u = e^x etc.) and also integration by parts but have had no luck.

As I said, I'm not sure if what I'm trying to do can be done, but if it can I'm hoping someone can see how to do it and suggest a productive approach.

Thanks in advance to all who have read this.