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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.

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Related:… – JavaMan Feb 10 '12 at 18:16
It's not difficult to manually apply the Liouville-Risch algorithm to this special case. For the few pages of theory needed see Rosenlicht's 1972 Monthly exposition Integration in finite terms. See also this answer. – Math Gems Feb 10 '12 at 20:03
Thank you ben crowell, for giving me a chance to answer. – Iyengar Feb 11 '12 at 17:39
@MathGems : Even though your links are good, but I think that we need to use the consequence of the actual theorem instead of heading towards it directly ( for the sake of ease ). – Iyengar Feb 11 '12 at 17:50
@Ben The accepted answer is not correct, nor can it be fixed - see my comment there. – Math Gems Feb 11 '12 at 18:11

See this sci.math article from 1993:

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