It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative.


The proof I know goes as follows: 
Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension for a suitable homogeneous differential equation for which $f$ is a solution.
Then one may calculate $G(E/F)$ and show it is connected and not abelian.


On the other hand, a calculation shows that if $K$ is a differential field extension of $F$ generated by elementary functions then the connected component of $G(K/F)$ is abelian, so it is impossible for an anti-derivative of $f$ to be contained in such a field $K$.


However, in classical Galois theory we can do much better, there, we know that a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable.


So to my question - is an analog of this is available in differential Galois theory? Is there a general method to determine by properties of $G(F/E)$ if $F$ is contained in a field of elementary functions?



Solvability in differential Galois theory
The analogue to "solvable by radicals" in differential Galois theory is "solvable by quadratures". The theorem says that a PV-extension is Liouvillian (adjoining primitives and exponentials) iff the connected component of the differential Galois group is solvable. See ["A first look at differential algebra"](http://www.math.cornell.edu/~hubbard/diffalg1.pdf) by Hubbard and Lundell, for an expository account.




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I slightly misread the question at first, thinking you were looking for the analogue of solvbility by radicals in differential algebra. When it comes to determining if the primitive of a function is elementary or not, the characterization is given by [Liouville's theorem](http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)). Now, for the general case of differential equations solvable in terms of elementary functions, there is a generalization of Liouville's theorem, that you can find in the article "Elementary and Liouvillian Solutions of Linear Differential Equations", by M.F. Singer and J. Davenport (link to Singer's papers [here](http://www4.ncsu.edu/~singer/ms_papers.html)).


