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How do I prove that the exponential integral $$f(x)=\int \frac{e^x}{x}\mathrm dx$$ is not an elementary function?

Also, what are the general methods and tricks to prove that an integral or solution to an equation is not an elementary function?

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    The general method is the *Differential Galois Theory* or the Galois theory of differential equations. There is an extremely nice exposition of the theory in a book by Andy Majid published by the AMS---at the end, IIRC, there is a complete argument that $\int e^{x^2}$ is not elementary, as an example.2011-10-28
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    Risch's services would be of great assistance.2011-10-28
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    http://en.wikipedia.org/wiki/Risch_algorithm2011-10-28

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