How difficult exactly is $\int\tan(x^2)\ dx$? How difficult exactly is $\int\tan(x^2)\ dx$ ? 

Is it possible to express this integral in terms of elementary functions? 

If not, is there anything one could say about it, that would be in some way helpful? 

I have not done anything to answer this question myself. (Well, I googled it, 
Wolfram alpha tells me no result found in terms of standard mathematical functions, so it seems safe to assume that no such result exist.) 

This integral looks somewhat similar to $\int e^{x^2} dx$ (which cannot be expressed in terms of elementary functions) but I just need some reassurance (possibly with a link or an explanation) specifically for $\int\tan(x^2)\ dx$ .  

Just in case, here is the Taylor series expansion
$\tan(x) = x+x^3/3+2x^5/15+17x^7/315+62x^9/2835+O(x^{11})$ and
$\tan(x) = \sum_{n=0}^\infty \dfrac{(-1)^{(n-1)}2^{2n}(2^{2n}-1) B(2n)}{(2n)!} x^{2n-1}$, where $B(n)$ are the Bernoulli numbers.
Someone asked me about this integral and I realized I couldn't say much about it.
If the question is : "Is it possible to express the integral $\int \tan(x^2)dx$  in terms of elementary functions ?" the answer is : Yes, on the form of infinite series of elementary functions.

If the question is : "Is it possible to express the integral $\int \tan(x^2)dx$  in terms of the combination of a finite number of elementary functions ?" the answer is : No. (as it was already pointed out in a preceeding answer).

If the question is : "How difficult exactly is $\int \tan(x^2)dx$ ?" the answer is : No more difficut than the integrals : $$\int \sin(x^2)dx=\sqrt{\frac{\pi}{2}}\  S
\left( \sqrt{\frac{2}{\pi}}\ x\right)+constant$$
where $S(X)$ is defined as a special function, namely the Fresnel S integral : http://mathworld.wolfram.com/SineIntegral.html

and no more difficult than the integral : $$\int \cos(x^2)dx=\sqrt{\frac{\pi}{2}}\  C
\left( \sqrt{\frac{2}{\pi}}\ x\right)+constant$$
where $C(X)$ is defined as a special function, namely the Fresnel C integral: http://mathworld.wolfram.com/CosineIntegral.html

The only difference is that in : $$\int \tan(x^2)dx=\sqrt{\frac{\pi}{2}}\  T
\left( \sqrt{\frac{2}{\pi}}\ x\right)+constant$$
the special function $T(X)$ is not referenced among the standard special functions, doesn't appear in the handbooks of special functions and is not implemented in the maths softwares.

One could say that just giving a name to an integral is no more than a cleaver trick. Nevertheless, one should think about it. A paper for general public on the subject : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
Impossible to express in terms of elementary functions. The proof is quite clever, and you should give it a shot. If you can't figure it out, let me know and I'll post the solution.

Here's a hint: http://en.wikipedia.org/wiki/Combinatorial_species#Further_operations