Proof that $\int x^x dx$ can't be done in terms of elementary functions?

Is there any easy proof that $\int x^x dx$ can't be done in terms of elementary functions? I know this is true, because the Risch algorithm gives a decision process for integration in terms of elementary functions, Axiom provides a complete software implementation of the Risch algorithm, and Axiom can't do the integral. However, it would be nicer to have a human-readable proof. If it could be reduced to a standard special function such as a hypergeometric function, then we could reduce the proof to a proof that that function can't be expressed in terms of elementary functions. But neither Axiom nor Wolfram Alpha can reduce it to any other form.

Solution 1:

See this sci.math article from 1993: http://groups.google.com/group/sci.math/msg/6b68b00362baf65f