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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.

Mirko
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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

JJacquelin
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  • I suppose that the second one is for $\cos(x^2)$ – Claude Leibovici Dec 03 '14 at 09:43
  • Of course, it's a typo. Merci, Claude ! – JJacquelin Dec 03 '14 at 10:25
  • Merci, the following answer of yours seems to be related http://math.stackexchange.com/questions/917830/very-few-elementary-functions-with-elementary-antiderivatives – Mirko Dec 03 '14 at 13:16
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    The following question may also be considered generally related What makes elementary functions elementary? – Mirko Dec 03 '14 at 13:26
  • FWIW, I like symmetry and completeness, thus I think $T(x)$ should exist, and also $\mathrm{Se}(x)$, $\mathrm{Cs}(x)$ and $\mathrm{Ct}(x)$ (guess). Likewise also for the "divided by $x$" versions: there should be $\mathrm{Ti}(x)$, $\mathrm{Sei}(x)$, $\mathrm{Csi}(x)$, and $\mathrm{Cti}(x)$. (You can guess). Rounding these out to a complete matched set is just satisfying. We have 6 standard trig functions, so there should be 6 in each set of corresponding simple non-elementary integrals. – The_Sympathizer Nov 10 '18 at 14:04
  • @ The_Sympathizer. Well, it is satisfying to define new special functions with elegant relationships. But it's far to be sufficient. A new special function has to be supported with a bunch of studies, properties, relationship with other special functions, etc. All this background must be published and made available in the math littérature so that, if someone refer to the new special function every one can find the background without having to reinvent the whole. I am afraid that $T(x) , Se(x) , Cs(x) , Ct(x) , Sei(x) , Csi(x) , Cti(x) remain wishful thinking for long. But why not ? – JJacquelin Nov 10 '18 at 17:12
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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 $\mathbb{R}$

doraemonpaul
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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

Aditya Hase
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  • Thank you, your hint just makes something that looks mysterious to me to look even more mysterious, I have no idea how to relate your hint to this problem (did you post the wrong link ? :) – Mirko Dec 03 '14 at 02:14
  • If you post an answer, post the answer. – Alec Dec 03 '14 at 10:32