You can't express it in elementary functions.
Actually it's not difficult. It's just that nobody bothered to think of defining a special function to account for its antiderivative.
There is a solution - the function F such that dF/dx is x^x. This function is also known as the Fantastic F-function.
It can't be written anymore nicely than that. There is nothing surprising about this. Almost no functions have integrals that can be written out nicely and explicitly in some closed form.
See this for elementary functions:
Elementary function - Wikipedia, the free encyclopedia