What features of elementary functions define a class of functions whose consecutive indefinite integration also gives an elementary function?

Is there a way to check whether a given elementary function has such property?

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What features of elementary functions define a class of functions whose consecutive indefinite integration also gives an elementary function? Is there a way to check whether a given elementary function has such property? |
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e.g.$x\mapsto \exp (x^2)$. – Loïc Teyssier May 3 at 8:00