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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?

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marked as duplicate by Rahul, Hans Lundmark, Chris Eagle, Henning Makholm, J. M. Aug 25 '12 at 12:02

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

You may be interested in this theorem by Liouville:… – Romeo Aug 25 '12 at 8:24
up vote 3 down vote accepted

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.

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