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

Question

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.

Answer 1

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

Answer 2

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

Answer 3

Please note that the no-closed-form integrals in fact can be further classified as four types:

Type $1$: Can be expressed as infinite series whose its radius of convergence covers on $\mathbb{C}$ or $\mathbb{R}$

Type $2$: Can only be expressed as infinite series whose its radius of convergence covers on finite ranges, to get the results suitable on $\mathbb{C}$ or $\mathbb{R}$ should have cases divisions

Type $3$: Other than Type $1$ and Type $2$ but the software can express them as known special functions, we can only follow the expressions from the software

Type $4$: Other than Type $1$ and Type $2$ and even the software cannot express them as known special functions, if we really want to force to solve them we should only use the formula e.g. this one.

Unfortunately, $\int\tan x^2~dx$ is in fact belongs to Type $4$ because of the following reasons:

$1.$ Wolfram fails to solve this integral.

$2.$ The radius of convergence of the power series of $\tan x$ is only $\dfrac{\pi}{2}$ and its coefficients have no-closed-form, and we should have infinitely many times of cases divisions in order to get the results suitable on $\mathbb{C}$ or $\mathbb{R}$