# Examples of non-trivial exclusively irrational integrals?

One very famous integral is

$$\int_{\mathbb{R}} \frac{\cos(x)}{x^2 +1} \, dx = \frac{\pi}{e} \tag{1}$$

as is shown in the answers to this question.

I find this integral particularly interesting as the result is written exclusively as a combination (by "combination" I mean a product/quotient/addition/exponentiation/logarithm) of irrational numbers, where I'm using "exclusively irrational" here to mean that the answer doesn't involve other factors of rational numbers combined with the irrationals. For example, the integral:

$$\int_{0}^{\infty} \frac{x^2}{e^x-1}\, dx = 2 \zeta(3)$$

I would not consider being "exclusively" irrational because of the factor of $$2$$ multiplying $$\zeta(3)$$.

I decided to look for other exclusively irrational integrals similar to $$(1)$$ which combine several irrational numbers in their result, but to my surprise, I couldn't find many examples similar to this. Most of the results I found where "single-irrational", like the following integrals:

$$\int_{\mathbb{R}} e^{-x^2}\, dx = \sqrt{\pi}, \qquad \int_{0}^{1} \ln\left(\ln\left(\frac{1}{x}\right)\right)\, dx=-\gamma, \qquad \int_{1}^{\infty}\frac{\ln(x)}{1+x^2} \, dx = G$$

which, although they are exclusively irrational, they can also be written in terms of a single famous irrational (hence the moniker I gave them). Some other common finds were "near-misses" like:

$$\int _0^{\infty }e^{-x}\ln ^2\left(x\right)\ dx = \gamma^2 + \frac{\pi^2}{6}, \qquad \int_{1}^{\infty} \frac{(x^4 - 6x^2+1)\ln(\ln(x))}{(1+x^2)^3}\, dx = \frac{2G}{\pi}$$

In fact, the only other exclusively irrational integral which wasn't also a single-irrational that I found was the integral

$$\int_{0}^{\infty} \frac{(1-x^2) \, \text{sech}^2\left(\frac{\pi x}{2} \right)}{(1+x^2)^2}\, dx = \frac{\zeta(3)}{\pi}\tag{2}$$

Of course, there are trivial integrals that indeed give exclusively irrational results. For example

$$\int_{0}^{\frac{\pi}{e}} 1 \, dx = \frac{\pi}{e}$$

but I would like to avoid these types of integrals. Another type is the "disguised" solution, which would be something like

$$\int_{\mathbb{R}} \frac{\sin(x)}{\color{purple}{e}x}\, dx= \frac{\pi}{e}, \qquad \int_{-1}^1\frac{1}{\color{purple}{4}x}\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ dx =\pi \, \text{arccot}\left(\sqrt{\varphi} \right)$$

which in reality are just single-irrational solutions or near-misses where we just multiplied a $$\color{purple}{\text{factor}}$$ on both sides. I would also like to avoid these types of integrals.

My question is:

Does anyone know any exclusively irrational integrals like $$(1)$$ and $$(2)$$ where you combine several different irrationals in the result? Preferably avoiding single-irrational, disguised, or trivial integrals like my other examples.

Ideally I would like results that exclusively combine irrational (and also very likely but still unproven to be irrational) numbers such as $$e,\,\pi$$ , Golden ratio $$\varphi ,\, \zeta(\text{Odd integer}),\,\ln(\text{Prime number}),\, \sqrt{\text{Prime number}}$$, Euler-Mascheroni constant and Catalan's constant; Where by "combination" I mean that these numbers are being added/multiplied/divided/exponentiated or being the argument of a trig function, in a way that doesn't simplify to factors of rational numbers, i.e. without something like $$\ln\left(e^2\right)$$.

Any help or suggestions are greatly appreciated. Thank you very much!

• I think that the $big-list tag is more relevant here than the integration or the recreational-mathematics tags. Sep 16, 2021 at 16:35 • Hmmm, it's interesting, how do you know that Euler–Mascheroni constant$\gamma\$ is irrational? Sep 16, 2021 at 16:38
• @Azlif, ahhhhh, indeed my wishful thinking once again made an appearance. I forget how many likely candidates have not been proven to be irrational. Thanks for the comments! Sep 16, 2021 at 16:51
• $$\int_0^\infty \exp\left(-\frac{3x^2+15}{2x^2+18}\right)\cos\left(\frac{2x}{x^2+9}\right)\frac{dx}{x^2+1}=\frac{\pi}{e}$$ $$\int_0^1 x^{-x}(1-x)^{x-1}\sin\pi x dx=\frac{\pi}{e}$$ $$\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}=\pi G$$ Sep 16, 2021 at 17:12
• Most of them are already posted on this site, that's why I posted them as a comment. Sep 16, 2021 at 17:56

This one is by-no-means trivial $$\int_0^1 \frac{\arctan^2x\ln\frac{x}{(1-x)^2}}xdx=G^2$$

• Obviously. This is the first time I saw Catalan constant squared in the value of an integral. Sep 16, 2021 at 18:27
• @LaxmiNarayanBhandari - Constructed it myself. Sep 18, 2021 at 23:58
• here are few more subtle ones \begin{align} \ln\frac2\pi &=\int_0^1 \frac{1-x}{(1+x)\ln x}dx\\ -\gamma \pi&= \int_{\mathbb R}\frac{\sin x\ln|x|}xdx\\ \frac\pi{\phi^{1/2}}&=\int_0^\pi \frac{\tan^{-1}(1+\cos x)}{1+\cos x}dx\\ \end{align} Sep 18, 2021 at 23:59
• OMG. You guys construct integrals. 😳 Sep 19, 2021 at 3:18
• @Quanto, in case you wish to share the method you used to construct this result, I started a bounty in this question for alternative proofs of this problem. I'm very curious to know how one comes up with a result like this! Mar 19, 2022 at 0:05

Whilst looking on this site for the integrals @Zacky posted in the comments, I found some more results:

• From this answer by Franklin Pezzuti Dyer

\begin{align*} \int_0^{\frac{\pi}{2}} \ln(x^2+\ln^2(\cos(x)))\, dx&=\pi\ln(\ln(2)) \end{align*}

• From this and this answer by Zacky (thanks again :D)

\begin{align*}\int_{\mathbb{R}} \frac{\sin \left(x-\frac{1}{x}\right) }{x+\frac{1}{x}}\, dx=\frac{\pi}{e^2}\\ \int_0^\frac{\pi}{2} x\ln\left(\cot\left(\frac{x}{2}\right)\left(\frac{\sec x}{2}\right)^4\right)\, dx=\pi G \end{align*}

• From this answer by Felix Marin

$$\int_{0}^{\frac{\pi}{2}} \ln \left(1+4\sin^4 (x)\right)\, dx = \pi \left(\ln \left( \varphi+\sqrt{\varphi} \right) - \ln(2)\right)$$

$$\int_{0}^{1} \frac{\ln\left(\ln^2\left(\frac{1}{x} \right) \right)}{(1 + x)^2} \, dx = \ln(\pi) - \ln(2) - \gamma$$
Here are some $$\pi$$ over $$e$$ variations on a theme. Obviously there is a close connection between each of the integrals. \eqalign{ \int_0^\infty \frac{\sin x}{x(1 + x^2)} dx &= \pi - \frac{\pi}{e};\cr \int_{-\infty}^\infty \frac{x \sin x}{1 + x^2} dx &= \frac{\pi}{e};\cr \int_{-\infty}^\infty \frac{x \sin (2x)}{(1 + x^2)^2} dx &= \frac{\pi}{e^2};\cr \int_0^\infty \frac{\cos (3x)}{(1 + x^2)^2} dx &= \frac{\pi}{e^3};\cr \int_0^\infty \frac{x\sin(4x)}{(1 + x^2)^2} dx &= \frac{\pi}{e^4}.\cr }
And for something a little different (and contrived but I think still within the bounds of your requirements): \eqalign{ \int_0^\infty \frac{\tanh (x^4) \operatorname{sech} (x^4)}{x} dx &= \frac{G}{\pi};\cr \int_0^\infty \frac{\tanh (x^7) \operatorname{sech}^2(x^7)}{x} \, dx &= \frac{\zeta (3)}{\pi^2}. }
Also: $$e$$ and $$\pi$$ are just as irrational as $$\frac{e}{2}$$, in fact, our knowledge about the possibility irrationality of $$e + \pi$$ and $$e\pi$$ is just slightly more than of $$4\gamma$$'s possible irrationality, a number which, for all we know, could itself be a multiple of $$\ln(\zeta(3))$$.
I think a similar question with requirements that are slightly less arbitrary is: What are some examples of irrational or transcendental numbers being periods or numbers that are irrational or transcendental to $$\Bbb{Q}[\pi]$$ being "pseudoperiods" (which allow trig functions in the integrand and $$\pi$$ in the limits)?