Check if antiderivative is elementary

I was experimenting with various integration techniques, and I stumbled upon the integral:

$$ \int{\frac{\tan{x}}{x} dx} $$

I tried using a number of methods, but I could not solve it. Checking on WolframAlpha, I found that the integral does not have an elementary antiderivative. However, I am not satisfied with simply assuming it's non-elementary if I have difficulty in solving it.

My question: Is there a test to determine if an antiderivative is elementary or not?

Solution 1:

A Ph.D paper at MIT before the turn of the century presented an algorithm that can integrate any function that has an indefinite integral expressible in terms of only elemetary functions.

The last I had heard, Mathematica had not implemented that full algorithm, but I would not be surprised if it has by now. In that case, if Mathematica can't find the indefinite integral, then it does not exist in terms of elementary functions.

The situation for definite integrals is much more difficult, and no general results are known.