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