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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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This is surely described in algebraic terms in the work of Liouville on the non-integrability (in your sense) of e.g. $x\mapsto \exp (x^2)$. –  Loïc Teyssier May 3 at 8:00
    
@Loïc Teyssier well I mean infinite integrability, not just integrability. –  Anixx May 3 at 8:03
    
I understand, but surely the latter implies the former. Hence you need to impose infinitely many such known algebraic conditions. You might very well find an explicit algebraic criterion as a result. –  Loïc Teyssier May 3 at 8:24

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