There are an infinite number of functions out there, and you can put an integral sign in front of any of them. Some of those functions are pretty strange and/or ugly. There is no reason why such an integral should have a representation in the very limited set of elementary functions. In fact when you get such a representation, you could consider yourself lucky.
For practical purposes, any convergent integral can be evaluated to any desired degree of accuracy. There are numerous numeric methods available with which to do that. And even if an integral should have some elementary function representation, that most often is a lot harder to evaluate than just running a numeric method on the integral. And you wouldn't be more accurate if the answer were cos(28.34) and you had to evaluate that.
Similarly, if you are going to use the integral in a further computation, it might be easiest to leave it as an integral, rather than tangling yourself up in things that have a lot of terms like sec(log^{-1}(x^{2})) -- or much worse.
So why do calculus classes teach you to find anti-derivatives? First, if the problem is going to resolve itself into an easy anti-derivative you might as well use it. Second, to familiarize you better with the underlying concepts, such as the chain rule and the product rule of derivatives. Third to give you some experience with actual answers so you have some idea what the answer ought to look like. If your numeric method evaluates to 2034.86, and you know the answer can't be larger than 80, then you know you made a mistake in your computation.
Of course, nowadays, all those integrals can be accurately evaluated online; but still, you should have some knowledge of what kind of answer to expect and what it all means.
Best Answer
The derivative of an elementary function is an elementary function: the standard Calculus 1 differentiation methods can be used to find this derivative. So an antiderivative of a non-elementary function can't be elementary.
EDIT: More formally, by definition an elementary function is obtained from complex constants and the variable $x$ by a finite number of steps of the following forms:
To prove that the derivative of an elementary function, you can use induction on the number of these steps. In the induction step, suppose the result is true for elementary functions obtained in at most $n$ steps. If $f$ can be obtained in $n+1$ steps, the last being $f = f_1 + f_2$ where $f_1$ and $f_2$ each require at most $n$ steps, then $f' = f_1' + f_2'$ where $f_1'$ and $f_2'$ are elementary, and therefore $f'$ is elementary. Similarly for the other possibilities for the last step.