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How to solve $\int x^xdx$

Indefinite integral $$\int x^x dx $$ knowing that $$x^x=e^{x \ln x}$$ and then using method of partial integration recursively I can get an infinite series. Does exist a closed form of this integral expressed in terms of elementary functions. I am interested also where appear this sort of integral.

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Adi Dani
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The closed form for this integral doesn't exist. For proof, this would surely help: Proof that $\int x^x dx$ can't be done in terms of elementary functions?

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Aang
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