How can you prove that a function has no closed form integral?

It is a theorem of Liouville, reproven later with purely algebraic methods, that for rational functions f and g, g nonconstant, the antiderivative

[f(x)exp(g(x))]dx
can be expressed in terms of elementary functions if and only if there exists some rational function h such that it is a solution to the differential equation:

f=h+hg
ex2dx is another classic example of such a function with no elementary antiderivative.
I don't know how much math you've had, but some of this paper might be comprehensible in its broad strokes: http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf

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