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I have seen this claim on Wikipedia and other places. Which branch of mathematics does this result come from?

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This is a consequence of the Liouville-Ritt theory of integration in finite terms. You can find a brief sketch and references in my post here.

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I am not aware of any easy proof of this. Here is a proof that $\int e^{x^2}$ cannot be expressed in terms of elementary functions: http://math.hunter.cuny.edu/ksda/papers/rick_09_02.pdf

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Differential Galois theory.

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