how do we know that integral is non-elementary?

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How can you prove that a function has no closed form integral?

Is there a condition that states that the indefinite integration is non-elementary?

You may be interested in this theorem by Liouville: http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28differential_algebra%29 — Romeo, Aug 25 '12 at 08:24

score 3 · Accepted Answer · answered Aug 25 '12 at 08:42

There is a decision procedure called the Risch algorithm that will either tell you that the intergral is non-elementary, or produce an elementary anti-derivative. It is not an easy algorithm to execute, or even implement in a computer algebra system (although the latter has been done), so there is no hope of finding an easy condition for the existence of an anti-derivative.

how do we know that integral is non-elementary?

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