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while trying to evaluate an infinite sum, I came across this integral: $$ \displaystyle \mathcal{I}=\int_{0}^{\infty}\frac{\sin(x)}{x}\DeclareMathOperator{\dilog}{Li_{2}}\dilog\left(-e^{-x}\right)\mathrm dx\tag*{}$$ Its approximate value by Wolfram is: $$\approx -0.694981$$ This looks similar to a Feynman Trick integral, so I'm wondering if there's a possible parameter we could use to simplify this integral somehow. Honestly, closed-form is going to be a very lax term in regards to this integral. I think a possible solution could be in terms of Gamma functions of complex arguments, or similarly a solution viz. Stieltjes constants (generalized Stieltjes more likely). I'm curious to see what you guys have in mind.

Notice, this integral was derived as part of an expression for a solution to the following infinite series: $$\displaystyle\sum\limits_{k\geq 1}\frac{(-1)^{k}\arctan(k)}{k^{2}}=-\frac{\pi}{4}\zeta(2)-\mathcal{I}$$ If you'd like to tackle this infinite sum as well, go right ahead! It's pretty interesting. One thing I haven't tried so far but could be of use is the fact that $$(-1)^{k}=\cos(\pi k)$$ and possibly differentiate the sum and see if I could build either a differential equation or find a simpler form with a closed-form solution that could then be integrated back. Good Luck to you and thank you in advance for taking the time to explore this integral and sum with me!

Brian Constantinescu
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  • Any particular reason why the sum doesn't start at $k=1$? – Simply Beautiful Art Aug 11 '19 at 21:34
  • @Simply Beautiful Art This is how the sum was originally presented to me. I agree that it's just an added nuisance, so I will shift the index for this sum as it bares no conflict to its convergence. – Brian Constantinescu Aug 11 '19 at 21:55
  • Well, I differentiated the sum and got some silly $$\int_0^1\frac{\sin(\frac{4\pi x^2}{1+x^2})}{\cos(\frac{4\pi x^2}{1+x^2})-\cosh(\frac{4\pi x}{1+x^2})}+\dots~\mathrm dx$$ so... is this supposed to be solvable? – Simply Beautiful Art Aug 11 '19 at 22:19
  • It could very well be a lost cause. I hoped maybe we could express this sum at least in terms of some existing transcendental functions. There was [this post](https://math.stackexchange.com/questions/364452/evaluate-int-0-frac-pi2-frac11x21-tan-x-mathrm-dx/1381569#1381569) by Olivier Oloa in which he devised the notion of "Poly-Stieltjes coefficients" which can be used to find a closed-form for this sum in terms of those coefficients. But they are a fairly new thing, can't find any other literature on them. So I wondered if the solution could be formed in terms of more popular functions. – Brian Constantinescu Aug 11 '19 at 22:38

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