Prove $\int_{0}^{\infty}{1\over x}\cdot{1-e^{-\phi{x}}\over 1+e^{\phi{x}}}dx=\ln\left({\pi\over 2}\right)$

Hint:

First, set $\phi{x}\mapsto x$. We have

\begin{equation} I(a):=\int_{0}^{\infty}{1\over x}\cdot{1-e^{-{x}}\over 1+e^{{x}}}dx=\int_{0}^{\infty}{1\over x}\cdot{e^{-{x}}-e^{-2{x}}\over 1+e^{-{x}}}dx \end{equation}

Consider the parametric integral

\begin{equation} I(a):=\int_{0}^{\infty}{e^{-(a-1)x}\over x}\cdot{e^{-{x}}-e^{-2{x}}\over 1+e^{-{x}}}dx \end{equation}

and

\begin{equation} I'(a)=\int_{0}^{\infty}{{e^{-(a+1)x}-e^{-ax}}\over 1+e^{-{x}}}dx \end{equation}

Now use the geometric expansion

\begin{equation} {1\over 1+e^{-{x}}}=\sum_{k=0}^\infty\,(-1)^ke^{-k{x}} \end{equation}

and the following relation

\begin{equation} \sum_{k=0}^\infty\frac{(-1)^k}{(z+k)^{m+1}}=\frac1{(-2)^{m+1}m!}\!\left(\psi_m\left(\frac{z}{2}\right)-\psi_m\!\left(\frac{z+1}{2}\right)\right) \end{equation} then do as shown in this answer.

By Frullani's theorem:

$$ I = \sum_{n\geq 1}(-1)^{n+1}\int_{0}^{+\infty}\frac{e^{-n\varphi x}-e^{-(n+1)\varphi x}}{x}\,dx =\sum_{n\geq 1}(-1)^{n+1}\log\left(\frac{n+1}{n}\right)\tag{1}$$ and the RHS is the logarithm of Wallis' product, hence $\color{red}{\log\frac{\pi}{2}}$.

You may also notice that the $\varphi$ constant is irrelevant here, since it can be eliminated by the substitution $x=\frac{z}{\varphi}$ in the original integral.

$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{\mathrm{i}} \newcommand{\iff}{\Leftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\, #2 \,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ It's clear that the integral is $\ds{\phi}$-independent. ${\phi \equiv {1 + \root{5} \over 2} > 0}$.

Namely, $\ds{\int_{0}^{\infty}{1 - \expo{-\phi x} \over 1 + \expo{\phi x}} \,{\dd x \over x}\ \stackrel{\phi x\ \to\ x}{=}\ \int_{0}^{\infty}{1 - \expo{-x} \over 1 + \expo{x}}\,{\dd x \over x}}$

---

\begin{align} \int_{0}^{\infty}{1 - \expo{-x} \over 1 + \expo{x}}\,{\dd x \over x} & = \int_{0}^{\infty}{1 - \expo{-x} \over 1 + \expo{-x}}\,\expo{-x}\,{\dd x \over x} \ \stackrel{\expo{-x}\ =\ t}{=}\ -\int_{0}^{1}{1 - t \over 1 + t}\,{\dd t \over \ln\pars{t}} \\[3mm] & = \int_{0}^{1}{1 - t \over 1 + t}\ \overbrace{\int_{0}^{\infty}t^{\mu}\,\dd\mu}^{\ds{-\,{1 \over \ln\pars{t}}}}\ \,\dd t = \int_{0}^{\infty}\int_{0}^{1}{t^{\mu} - t^{\mu + 1} \over 1 + t}\,\dd\mu\,\dd t \\[3mm] & = \int_{0}^{\infty}\bracks{% 2\int_{0}^{1}{t^{\mu} - t^{\mu + 1} \over 1 - t^{2}}\,\dd\mu - \int_{0}^{1}{t^{\mu} - t^{\mu + 1} \over 1 - t}\,\dd\mu} \\[3mm] & = \int_{0}^{\infty}\bracks{% \int_{0}^{1}{t^{\mu/2 - 1/2} - t^{\mu/2} \over 1 - t}\,\dd\mu - \int_{0}^{1}{t^{\mu} - t^{\mu + 1} \over 1 - t}\,\dd\mu} \\[8mm] & = \int_{0}^{\infty}\left\lbrack% \int_{0}^{1}{1 - t^{\mu/2} \over 1 - t}\,\dd\mu - \int_{0}^{1}{1 - t^{\mu/2 - 1/2} \over 1 - t}\,\dd\mu\right. \\[3mm] & \left.\mbox{} + \int_{0}^{1}{1 - t^{\mu} \over 1 - t}\,\dd\mu - \int_{0}^{1}{1 - t^{\mu + 1} \over 1 - t}\,\dd\mu\right\rbrack \\[8mm] & = \int_{0}^{\infty}\bracks{% \Psi\pars{1 + {\mu \over 2}} - \Psi\pars{\half + {\mu \over 2}} + \Psi\pars{1 + \mu} - \Psi\pars{2 + \mu}}\,\dd\mu \\[3mm] & = \left.% \ln\pars{\Gamma^{2}\pars{1 + \mu/2}\Gamma\pars{1 + \mu} \over \Gamma^{2}\pars{1/2 + \mu/2}\Gamma\pars{2 + \mu}} \right\vert_{\ 0}^{\ \infty} \\[3mm] & = \underbrace{% \lim_{\mu \to \infty}\ln\pars{\Gamma^{2}\pars{1 + \mu/2} \over \Gamma^{2}\pars{1/2 + \mu/2}\pars{1 + \mu}}}_{\ds{-\ln\pars{2}}}\ -\ \underbrace{\ln\pars{\Gamma^{2}\pars{1}\Gamma\pars{1} \over \Gamma^{2}\pars{1/2}\Gamma\pars{2}}}_{\ds{-\ln\pars{\pi}}}\ =\ \color{#f00}{\ln\pars{\pi \over 2}} \end{align}

where $\Psi$ is the Digamma function and, by definition, $\ds{\Psi\pars{z} = \totald{\ln\pars{\Gamma\pars{z}}}{z}}$. In the above calculation we used the well known identities ( $\gamma$ is the Euler-Mascheroni constant ): \begin{align} \Psi\pars{z} + \gamma & = \int_{0}^{1}{1 - t^{z - 1} \over 1 - t}\,\dd t\,,\qquad\Re\pars{z} > 0 \\[3mm] \Gamma\pars{1} & = \Gamma\pars{2} = 1\,,\quad\Gamma\pars{\half} = \root{\pi} \,,\quad\Gamma\pars{z + 1} = z\,\Gamma\pars{z} \end{align}

The last $\ds{\mu \to \infty}$ limit can be evaluated with Stirling Asymptotic Formula.