Take the 2-minute tour ×  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?  share|improve this question 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. –  Mariano
Suárez-Alvarez Oct 28 '11 at 11:21 Risch's services would be of
great assistance. –  Ｊ. Ｍ. Oct 28 '11 at 11:21
en.wikipedia.org/wiki/Risch_algorithm –  pedja Oct 28 '11 at
11:23  1 Answer 1  All we need for this is a theorem of
Liouville (1835): Suppose that $f$ and $g$ are rational
functions with $f\neq 0$ and $g$ non-constant. Then $$\int f(x)
e^{g(x)} dx $$ is an elementary function if and only if there ex