Remarkable integral: $\int_0^{\infty} x \left(1 - \frac{\sinh x}{\cosh x-\sqrt 3/2} \right) \mathrm dx= -\frac{13 \pi ^2}{72}$?

Numerical evidence suggests that

$$\int_0^{\infty} x \left(1 - \frac{\sinh x}{\cosh x-\sqrt 3/2} \right) \mathrm dx= -\frac{13 \pi ^2}{72}$$

How can we prove this? I could not find a nice contour in the complex plane to integrate around. Integration by parts also didn't help. Mathematica finds a very complicated antiderivative in terms of special functions, but this was a contest problem so there must be a 'human' way to calculate it.

As O.L. helpfully pointed out, I had the sign wrong. It is corrected now.

• Would $x=\ln v$ be of benefit here? May 4, 2014 at 12:26
• Conjecture: $$\int^\infty_0 x\left(1-\frac{\sinh x}{\cosh x+\cos a\pi}\right)\stackrel?=\pi^2(\frac16-\frac{a^2}{2}).$$ May 4, 2014 at 13:12
• In general $$\int_{0}^{\infty} x \left(1- \frac{\sinh x}{\cosh x - \cos a} \right) dx = \pi a- \frac{a^2}2-\frac{\pi^2}3$$ Apr 13 at 13:32

Integration by parts does help: as $$1-\frac{\sinh x}{\cosh x-\cos\gamma}=\left(-\ln\frac{\cosh x-\cos\gamma}{e^x/2}\right)',$$ after IbP and further change of variables $t=e^{-x}$ the integral transforms into $$\int_0^{1}\frac{\ln\left(1-e^{i\gamma}t\right)\left(1-e^{-i\gamma}t\right)}{t}dt=-\left[2\Re\operatorname{Li}_2\left(e^{i\gamma}t\right)\right]_{0}^{1}=-2\Re\operatorname{Li}_2\left(e^{i\gamma}\right),$$ with $\gamma=\frac{\pi}{6}$. Now to get the result it remains to use the formula $$\Re\,\mathrm{Li}_2\left(e^{i\gamma}\right)=\frac{\gamma^2}{4}-\frac{\pi\gamma}{2}+\frac{\pi^2}6,\qquad \gamma\in(0,2\pi).$$ This also yields the conjecture mentioned in the comments.

Added: We can also obtain the result pretending that we don't know anything about dilogarithms. Namely, differentiate the integral with respect to parameter $\gamma$: \begin{align} \frac{\partial}{\partial \gamma}\int_0^{1}\frac{\ln\left(1-e^{i\gamma}t\right)\left(1-e^{-i\gamma}t\right)}{t}dt&=-i\int_0^1\left[\frac{e^{i\gamma}}{1-e^{i\gamma}t}-\frac{e^{-i\gamma}}{1-e^{-i\gamma}t}\right]dt=\\&=i\biggl[\ln\frac{1-e^{i\gamma}t}{1-e^{-i\gamma}t}\biggr]_0^1=-2 \biggl[\operatorname{arg}(1-e^{i\gamma}t)\biggr]_0^1=\\&=-2\biggl[\left(\frac{\gamma}{2}-\frac{\pi}{2}\right)-0\biggr]=\pi-\gamma, \end{align} where we again assume that $\gamma\in(0,2\pi)$. We can now integrate back with respect to $\gamma$ to get the previously obtained formula using that for $\gamma=0$ our integral reduces to computation of $\zeta(2)$ (expand the integrand into Taylor series w.r.t. $t$).

• Could you sketch or give a reference on how one can prove the formula for $\Re \mathrm{Li}_2(e^{i\gamma})$? May 4, 2014 at 13:48
• @user111187 I have added a kind of derivation. Otherwise this is basically the dilogarithm identity from here with $z$ on the unit circle. This can be proved analogously (differentiate to get elementary functions and integrate back using $\zeta(2)$). May 4, 2014 at 14:05
• Very nice, I appreciate your work. May 4, 2014 at 14:12
• Nice solution +1 May 4, 2014 at 18:47

The following is another approach that also involves differentiating under the integral sign.

Let $$I(\theta) = \int_{0}^{\infty} x \left(1- \frac{\sinh x}{\cosh x - \cos \theta} \right) \, \mathrm dx \, , \quad 0 < \theta < \pi.$$

Then $$I'(\theta) = \sin \theta \int_{0}^{\infty} \frac{x \sinh x}{\left(\cosh x - \cos \theta \right)^{2}} \, \mathrm dx.$$

I tried evaluating $$I'(\theta)$$ using contour integration, but calculating the residues became too tedious.

Instead let $$a$$ be a positive parameter, and let $$J(\alpha) = \int_{0}^{\infty}\frac{\mathrm d x}{\cosh (\alpha x)- \cos \theta} = \frac{1}{\alpha} \int_{0}^{\infty} \frac{du}{\cosh u - \cos \theta}.$$

Then $$I'(\theta) = -\sin (\theta) \, J'(1)$$.

To evaluate $$J(\alpha)$$, let's integrate the function $$f(z) = \frac{z}{\cosh z - \cos \theta}$$ around a rectangle contour in the upper half of the complex plane of height $$2 \pi i$$.

We get $$- 2 \pi i \int_{-\infty}^{\infty} \frac{dt}{\cosh t - \cos \theta} = 2 \pi i \, \left(\operatorname{Res} \left[f(z), i \theta \right]+ \operatorname{Res} \left[f(z), i \left(2 \pi - \theta\right)\right] \right)= 4 \pi i \, \frac{\left(\theta - \pi\right)}{\sin \theta}.$$

Therefore, $$J(\alpha) = \frac{1}{\alpha}\frac{\pi - \theta}{\sin \theta}$$, which means $$I'(\theta) =\pi - \theta$$.

Integrating with respect to $$\theta$$, we get $$I(\theta) = \pi \theta -\frac{\theta^{2}}{2} + C.$$

To determine the constant of integration, we can evaluate $$I(\theta)$$ at $$\theta = \frac{\pi}{2}$$.

\begin{align} I(\pi/2) &= \int_{0}^{\infty} x \left(1- \tanh x \right) \mathrm dx \\ &= (1- \tanh x) \frac{x^{2}}{2}\Bigg|^{\infty}_{0} + \frac{1}{2}\int_{0}^{\infty}\frac{x^{2}}{\cosh^{2} x} \mathrm dx \\ &= \frac{1}{2} \int_{0}^{\infty} \frac{x^{2}}{\cosh^{2} x} \mathrm dx \\ &= 2 \int_{0}^{\infty}x^{2} \, \frac{e^{-2x}}{(1+e^{-2x})^{2}} \, \mathrm dx \\ &= 2 \int_{0}^{\infty}x^{2} \, \sum_{n=1}^{\infty} (-1)^{n-1} n e^{-2nx} \mathrm dx \\ &= 2 \sum_{n=1}^{\infty} (-1)^{n-1} n \int_{0}^{\infty} x^{2} e^{-2nx} \, \mathrm dx \\ &= \frac{1}{2} \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^{2}} \\ &= \frac{\pi^{2}}{24}. \end{align}

So $$C= \frac{\pi^{2}}{24} -\frac{\pi^2}{2} + \frac{\pi^{2}}{8} = -\frac{\pi^2}{3}$$, and, therefore, $$I(\theta) = \pi \theta -\frac{\theta^{2}}{2} - \frac{\pi^{2}}{3}.$$