Integration of x^x

[razrushil]

I can't say whether it is useful, but it most likely won't be if you can't write it in shorter notation. I've seen something like this elsewhere, but terms were just written out and there didn't seem to be much of a pattern, at least at first glance.
If your results can be generalized more it can be like an integral for general tetration. This would be more difficult, but it has to start somewhere.