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My integration skills are okay. I know most common techniques and currently trying to sharpen up by going over the book Interesting integrals.

However, I do struggle to identify when an expression has no antiderivative in terms of elementary functions and often find myself spending a long time on integrals only for Wolfram Alpha to tell me I’m snookered.

Are there techniques available to be able to identify such things other than by experience?

Rumplestillskin
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    This is not an easy problem... From the theoretical standpoint, you can have a look at Liouville's theorem. – mathcounterexamples.net Jul 25 '18 at 09:29
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    I only know there is an algorithm, called the Risch algorithm (https://en.wikipedia.org/wiki/Risch_algorithm), which can verify if a given function has got an anti-derivative or not. But it is a computer algorithm so I am not sure if you can compute it by hand by I think not. I guess to know some of the special functions concerning not integratable function combinations, like the Sinc integral of the integral logarithm, is a first step. – mrtaurho Jul 25 '18 at 09:30
  • Related: https://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral – Hans Lundmark Jul 25 '18 at 10:12

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