Recognizing that a function has no elementary antiderivative [duplicate]

Is there a method to check whether a function is integrable?

Of-course trying to solve it is one but some questions in integration may be so tricky that I don't get the correct method to start off with those problems. So, is there a method to find correctly whether a function is integrable?

Clarification: I am asking about indefinite integrals which have no elementary anti derivative.

Maybe you are asking for a procedure for determining, given an elementary function $f(x)$, whether there is an elementary function $F(x)$ such that $F'(x)=f(x)$.

With some caveats, there is such a procedure, called the Risch Algorithm. For some discussion of the algorithm, please see the linked Wikipedia article.

Part-implementations of the Risch Algorithm are a component of various symbolic integration programs.