reference request - how do we know that integral is non-elementary?

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

Answer

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.