Examples of non-trivial exclusively irrational integrals?

Question

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 $\zeta(3)$.

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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

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

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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.

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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!

Answer 1

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

Answer 2

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

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

• From this AOPS thread

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

Answer 3

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): $$\eqalign{ \int_0^\infty \frac{\tanh (x^4) \operatorname{sech} (x^4)}{x} dx &= \frac{G}{\pi};\cr \int_0^\infty \frac{\tanh (x^7) \operatorname{sech}^2(x^7)}{x} \, dx &= \frac{\zeta (3)}{\pi^2}. }$$

Answer 4

Although there are some cases where an integral will be "nontrivially exclusively irrational" as you put it, with enough manipulation, a lot of them could be shown to be "in disguise" (trig rules, u-subs, etc.) I would in fact conjecture that an algorithm can be made to generate integrals equal to a target answer through Product Rule, Chain Rule, and various known identities, after which U-sub and filtering could be used to properly get integrals that appear to the naked eye as being "nontrivially exclusively irrational."

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)?