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

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:

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:


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