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I have seen people deduce at a glance that a given integrand's antiderivative cannot be expressed in terms of elementary functions. How is this done?

Hisham
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    They might know already that certain forms don't have elementary antiderivatives, or they might recognize the integral as related to a special function. – aschepler Feb 22 '23 at 23:19
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    They recognize known forms. You might want to look at this question for a brief discussion related to $\int \exp (x^2), dx$, one of the first to be proven to be "unintegrable" in terms of elementary functions. – lulu Feb 22 '23 at 23:22
  • thank you for the replies. I looked at the linked question and skimmed through the article on liouville's theorem but I've never studied differential algebra before. Is there a procedure I can do to verify if an elementary function's antiderivative is elementary? – Hisham Feb 23 '23 at 01:51
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    afaik, Liouville's theorem is the only systematical result in that field. I have however never seen a practitioner using it. What I have seen a lot were the phenomenal skills of experienced people spotting the tricks. – Kurt G. Feb 23 '23 at 04:37

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