Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$
By the substitution $x = e^{-t}$, we find that
$$\begin{align*} \int_{0}^{1} \frac{(\log (1/x))^s}{1+x^2} \; dx &= \int_{0}^{\infty} \frac{t^s e^{-t}}{1 + e^{-2t}} \; dt \\ &= \int_{0}^{\infty} \sum_{n=0}^{\infty} (-1)^n t^s e^{-(2n+1)t} \; dt \\ &= \sum_{n=0}^{\infty} (-1)^n \, \frac{\Gamma(s+1)}{(2n+1)^{s+1}} \\ &= \Gamma(s+1)L(s+1, \chi_4), \end{align*}$$
where $L(s, \chi_4)$ is the Dirichlet L-function of the unique non-principal character $\chi_4$ to the modulus 4. Often it is denoted as $\beta(s)$ and called the Dirichlet beta function. Thus differentiating both sides with respect to $s$ and plugging $s = 0$, we obtain a representation formula
$$\int_{0}^{1} \frac{\log \log (1/x)}{1+x^2} \; dx = \psi_0(1) \beta(1) + \beta'(1),$$
and the problem reduces to find the value of $\beta(1)$ and $\beta'(1)$. Note that $\beta(1) = \frac{\pi}{4}$ is just the Gregory series. For $\beta'(1)$, we first notice that the following functional equation holds.
$$ \beta(s)=\left(\frac{\pi}{2}\right)^{s-1} \Gamma(1-s) \cos \left( \frac{\pi s}{2} \right)\,\beta(1-s). $$
This follows from the general theory of Dirichlet L-functions, and one can consult with any analytic number theory textbook to find its proof. Therefore it is sufficient to calculate $\beta'(0)$. For $0 < s$, we have
$$\begin{align*} -\beta'(s) &= \sum_{n=1}^{\infty} \left[ \frac{\log(4n+1)}{(4n+1)^s} - \frac{\log(4n-1)}{(4n-1)^s} \right] \\ &= \sum_{n=1}^{\infty} \frac{1}{(4n)^s} \left[ \log \left( \frac{4n+1}{4n-1} \right) - \frac{1}{2n} \right] + 2^{-2s-1}\zeta(s+1) \\ & \qquad + \sum_{n=1}^{\infty} \left( \frac{1}{(4n+1)^s} - \frac{1}{(4n)^s} \right) \log (4n+1) \\ & \qquad + \sum_{n=1}^{\infty} \left( \frac{1}{(4n)^s} - \frac{1}{(4n-1)^s} \right) \log (4n-1) \\ & =: A(s) + 2^{-2s-1}\zeta(s+1) + B(s) + C(s). \end{align*}$$
We first estimate $B(s)$. As $n \to \infty$, we have
$$ \log \left( \frac{4n}{4n+1} \right) = -\frac{1}{4n} + O\left( \frac{1}{n^2} \right), \quad \log \left( \frac{4n}{4n-1} \right) = \frac{1}{4n} + O\left( \frac{1}{n^2} \right). $$
Thus when $s \to 0$,
$$\begin{align*} B(s) &= \sum_{n=1}^{\infty} \frac{1}{(4n)^s} \left[ \exp\left( s \log \left( \frac{4n}{4n+1} \right) \right) - 1 \right] \left[ \log (4n) - \log \left(\frac{4n}{4n+1} \right) \right] \\ &= \sum_{n=1}^{\infty} \frac{1}{(4n)^s} \left[ - \frac{s}{4n} + O \left(\frac{s^2}{n^2} \right) \right] \left[ \log (4n) + O \left(\frac{1}{n} \right) \right] \\ &= -s 2^{-2s-2} \sum_{n=1}^{\infty} \frac{1}{n^{s+1}} \log (4n) + O(s) \\ &= s 2^{-2s-2} \left[ \zeta'(s+1) - \zeta(s+1) \log 4 \right] + O(s). \end{align*}$$
Similar consideration also shows that
$$ C(s) = s 2^{-2s-2} \left[ \zeta'(s+1) - \zeta(s+1) \log 4 \right] + O(s).$$
Thus we have
$$ 2^{-2s-1}\zeta(s+1) + B(s) + C(s) = 2^{-2s-1} \left[ \zeta(s+1) + s \zeta'(s+1) - s \zeta(s+1) \log 4 \right] + O(s). $$
But since
$$\zeta(1+s) = \frac{1}{s} + \gamma + O(s),$$
we have
$$ \lim_{s\downarrow 0} \left( 2^{-2s-1}\zeta(s+1) + B(s) + C(s) \right) = \frac{\gamma}{2} - \log 2.$$
For $A(s)$, the summands are positive with possible finite exceptional terms. Thus the Monotone Convergence Theorem guarantees that
$$ \lim_{s\downarrow 0} A(s) = \sum_{n=1}^{\infty} \left[ \log \left( \frac{4n+1}{4n-1} \right) - \frac{1}{2n} \right]. $$
Let $L$ denote this limit. Then by Stirling's formula,
$$\begin{align*} e^{L} & \stackrel{N\to\infty}{\sim} \prod_{n=1}^{N} \left( \frac{4n+1}{4n-1} \right) e^{-1/2n} \sim \frac{e^{-\gamma/2}}{\sqrt{N}} \prod_{n=1}^{N} \left( \frac{n+(1/4)}{n-(1/4)} \right) \\ & \sim \frac{e^{-\gamma/2}}{\sqrt{N}} \frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{5}{4}\right)} \frac{\Gamma\left(N+\frac{5}{4}\right)}{\Gamma\left(N+\frac{3}{4}\right)} \sim \frac{e^{-\gamma/2}}{\sqrt{N}} \frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{5}{4}\right)} \frac{\left( \frac{N + (5/4)}{e} \right)^{N+\frac{5}{4}}}{\left( \frac{N + (3/4)}{e} \right)^{N+\frac{3}{4}}} \\ & \sim e^{-\gamma/2} \frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{5}{4}\right)} = 4 e^{-\gamma/2} \frac{\pi \sqrt{2}}{\Gamma\left(\frac{1}{4}\right)^2}, \end{align*}$$
where we have used the Euler's reflection formula in the last line. Combining all the efforts, we obtain
$$-\beta'(0) = \log (2 \pi \sqrt{2}) - 2 \log \Gamma\left(\frac{1}{4}\right) .$$
Now taking logarithmic differntiation to the functional equation, we have
$$ \frac{\beta'(s)}{\beta(s)} = \log\left(\frac{\pi}{2}\right) - \psi_0 (1-s) - \frac{\pi}{2} \tan \left( \frac{\pi s}{2} \right) - \frac{\beta'(1-s)}{\beta(1-s)}. $$
Taking $s = 0$, we have
$$ \frac{\beta'(0)}{\beta(0)} = \log\left(\frac{\pi}{2}\right) + \gamma - \frac{\beta'(1)}{\beta(1)} \quad \Longrightarrow \quad \beta'(1) = \beta(1) \left[ \log\left(\frac{\pi}{2}\right) + \gamma - \frac{\beta'(0)}{\beta(0)} \right]. $$
But again by the functional equation, we have $\beta(0) = \frac{1}{2}$. Therefore
$$ \beta'(1) = \frac{\pi}{4} \left[ \gamma + 2 \log 2 + 3 \log \pi - 4 \log \Gamma\left(\frac{1}{4}\right) \right], $$
and hence
$$ \int_{0}^{1} \frac{\log \log (1/x)}{1+x^2} \; dx = \frac{\pi}{4} \left[ 2 \log 2 + 3 \log \pi - 4 \log \Gamma\left(\frac{1}{4}\right) \right], $$
which is identical to the proposed answer.
$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle #1 \right\rangle} \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace} \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left( #1 \right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\int_{0}^{1}\ln\pars{\ln\pars{1 \over x}}\,{\dd x \over 1 + x^{2}} ={\pi \over 2}\,\ln\pars{\root{\vphantom{\large A}2\pi}\, {\Gamma\pars{3/4} \over \Gamma\pars{1/4}}}:\ {\Large ?}}$
With $\ds{x \to 1/x}$: $$ \color{#f00}{\int_{0}^{1}\ln\pars{\ln\pars{1 \over x}}\,{\dd x \over 1 + x^{2}}}= \int_{\infty}^{1}\ln\pars{\ln\pars{x}}\,\pars{-\,{\dd x/x^{2} \over 1 + 1/x^{2}}} =\int_{1}^{\infty}{\ln\pars{\ln\pars{x}} \over 1 + x^{2}}\,\dd x $$
With $x \equiv \expo{t}\quad\iff\quad t = \ln\pars{x}$: \begin{align} &\color{#f00}{\int_{0}^{1}\ln\pars{\ln\pars{1 \over x}}\,{\dd x \over 1 + x^{2}}}= \int_{0}^{\infty}{\ln\pars{t} \over 1 + \expo{2t}}\,\expo{t}\dd t =\int_{0}^{\infty}\ln\pars{t}\expo{-t}\,{1 \over 1 + \expo{-2t}}\,\dd t \\[3mm]&=\int_{0}^{\infty}\ln\pars{t}\expo{-t}\, \sum_{\ell = 0}^{\infty}\pars{-1}^{\ell}\expo{-2\ell t}\,\dd t =\sum_{\ell = 0}^{\infty}\pars{-1}^{\ell}\lim_{\mu \to 0}\partiald{}{\mu}\bracks{% \int_{0}^{\infty}t^{\mu}\expo{-\pars{2\ell + 1}t}\,\dd t} \\[3mm]&=\sum_{\ell = 0}^{\infty}\pars{-1}^{\ell}\lim_{\mu \to 0}\partiald{}{\mu}\bracks{% {1 \over \pars{2\ell + 1}^{\mu + 1}}\ \overbrace{\int_{0}^{\infty}t^{\mu}\expo{-t}\,\dd t}^{\ds{\Gamma\pars{\mu + 1}}}\ } \end{align} where $\Gamma\pars{z}$ is the Gamma Function.
\begin{align} &\color{#f00}{\int_{0}^{1}\ln\pars{\ln\pars{1 \over x}}\,{\dd x \over 1 + x^{2}}}= \sum_{\ell = 0}^{\infty}\pars{-1}^{\ell}\lim_{\mu \to 0}\partiald{}{\mu}\bracks{% {\Gamma\pars{\mu + 1} \over \pars{2\ell + 1}^{\mu + 1}}} \\[3mm]&= \lim_{\mu \to 0}\partiald{}{\mu}\bracks{% \Gamma\pars{\mu + 1}\sum_{\ell = 0}^{\infty} {\pars{-1}^{\ell}\over \pars{2\ell + 1}^{\mu + 1}}} \\[3mm]&= \lim_{\mu \to 0}\braces{% \Gamma'\pars{\mu + 1}\sum_{\ell = 0}^{\infty} {\pars{-1}^{\ell}\over \pars{2\ell + 1}^{\mu + 1}} +\Gamma\pars{\mu + 1}\partiald{}{\mu}\bracks{% \sum_{\ell = 0}^{\infty}{\pars{-1}^{\ell}\over \pars{2\ell + 1}^{\mu + 1}}}} \\[3mm]&=-\gamma\sum_{\ell = 0}^{\infty}{\pars{-1}^{\ell}\over 2\ell + 1} + \lim_{\mu \to 0}\partiald{}{\mu} \sum_{\ell = 0}^{\infty}{\pars{-1}^{\ell}\over \pars{2\ell + 1}^{\mu + 1}}\tag{1} \end{align} In this result, we used $\Psi\pars{1} = -\gamma$ and $\Gamma\pars{1} = 1$ where $\Psi\pars{z} \equiv \dd\ln\Gamma\pars{z}/\dd z$ is the Digamma Function and $\gamma$ is the Euler-Mascheroni constant.
The first $\ell$-sum in the right member of $\pars{1}$ is given by: \begin{align} &\sum_{\ell = 0}{\pars{-}^{\ell} \over 2\ell + 1}= \sum_{\ell = 0}\pars{{1 \over 4\ell + 1} - {1 \over 4\ell + 3}} ={1 \over 8}\sum_{\ell = 0}{1 \over \pars{\ell + 1/4}\pars{\ell + 3/4}} \\[3mm]&=-\,{1 \over 4}\bracks{\Psi\pars{1 \over 4} - \Psi\pars{3 \over 4}} = {\pi \over 4} \end{align} where we used the identities: \begin{align} \sum_{\ell = 0}^{\infty}{1 \over \pars{\ell + z_{0}}\pars{\ell + z_{1}}} &={\Psi\pars{z_{0}} - \Psi\pars{z_{1}} \over z_{0} - z_{1}}\tag{1.1} \\[3mm]\Psi\pars{z} - \Psi\pars{1 - z} &= -\pi\cot\pars{\pi z}\tag{1.2} \end{align} $$ \mbox{Then,}\quad \color{#f00}{\int_{0}^{1}\ln\pars{\ln\pars{1 \over x}}\,{\dd x \over 1 + x^{2}}}= -\,{1 \over 4}\,\gamma\pi + \lim_{\mu \to 0}\partiald{}{\mu} \sum_{\ell = 0}^{\infty}{\pars{-1}^{\ell}\over \pars{2\ell + 1}^{\mu + 1}}\tag{2} $$ Also, \begin{align} \sum_{\ell = 0}^{\infty}{\pars{-1}^{\ell}\over \pars{2\ell + 1}^{\mu + 1}} &=\sum_{\ell = 0}^{\infty}{1 \over \bracks{2\pars{2\ell} + 1}^{\mu + 1}} -\sum_{\ell = 0}^{\infty}{1 \over \bracks{2\pars{2\ell + 1} + 1}^{\mu + 1}} \\[3mm]&=2^{-2\mu - 2}\bracks{% \sum_{\ell = 0}^{\infty}{1 \over \pars{\ell + 1/4}^{\mu + 1}} -\sum_{\ell = 0}^{\infty}{1 \over \pars{\ell + 3/4}^{\mu + 1}}} \\[3mm]&=2^{-2\mu - 2}\bracks{% \zeta\pars{\mu + 1,{1 \over 4}} - \zeta\pars{\mu + 1,{3 \over 4}}} \end{align} where $\ds{\zeta\pars{s,q} \equiv \sum_{n = 0}^{\infty}{1 \over \pars{q + n}^{s}}}$ with $\Re\pars{s} > 1$ and $\Re\pars{q} > 0$ i s the Hurwitz Zeta Function or/and Generalizated Zeta Funcion .
So, \begin{align} &\lim_{\mu \to 0}\partiald{}{\mu}\sum_{\ell = 0}^{\infty}{\pars{-1}^{\ell}\over \pars{2\ell + 1}^{\mu + 1}} \\[3mm]&=-\,{1 \over 4}\,\ln\pars{2}\ \underbrace{\overbrace{\sum_{\ell = 0}^{\infty}{1 \over \pars{\ell + 3/4}\pars{\ell + 1/4}}} ^{\ds{2\bracks{\Psi\pars{3/4} - \Psi\pars{1/4}} = 2\pi}}} _{\ds{\mbox{See}\ \pars{1.1}\ \mbox{and}\ \pars{1.2}}} +\ {1 \over 4}\ \overbrace{\partiald{}{\mu}\bracks{% \zeta\pars{\mu,{1 \over 4}} - \zeta\pars{\mu,{3 \over 4}}}_{\mu\ =\ 1}} ^{\ds{-\gamma_{1}\pars{1/4} + \gamma_{1}\pars{3/4}}} \end{align} where $\gamma_{n}\pars{z}$ is a Generalizated Stieltjes Constant .
With this result, $\pars{2}$ is reduced to: \begin{align} \color{#f00}{\int_{0}^{1}\ln\pars{\ln\pars{1 \over x}}\,{\dd x \over 1 + x^{2}}} &=-\,{1 \over 4}\,\braces{% \pi\bracks{% \gamma + 2\ln\pars{2}} + \gamma_{1}\pars{1 \over 4} - \gamma_{1}\pars{3 \over 4}} \tag{3} \end{align} The difference $\gamma_{1}\pars{1/4} - \gamma_{1}\pars{3/4}$ is evaluated with the 1846 Carl Malmsten identity : $$ \gamma_{1}\pars{m \over n} - \gamma_{1}\pars{1 - {m \over n}} =-\pi\bracks{\gamma + \ln\pars{2\pi n}}\cot\pars{m\pi \over n} + 2\pi\sum_{\ell = 1}^{n - 1} \sin\pars{{2\pi m \over n}\,\ell}\ln\pars{\Gamma\pars{\ell \over n}} $$
With $m = 1$ and $n = 4$: \begin{align} &\gamma_{1}\pars{1 \over 4} - \gamma_{1}\pars{3 \over 4} \\[3mm]&=-\pi\bracks{\gamma + \ln\pars{8\pi}}\cot\pars{\pi \over 4} + 2\pi\sum_{\ell = 1}^{3}\sin\pars{\pi\,\ell \over 2} \ln\pars{\Gamma\pars{\ell \over 4}} \\[3mm]&=-\pi\bracks{\gamma + 2\ln\pars{2} + \ln\pars{2\pi}} \\[3mm]&\phantom{=} + 2\pi\bracks{\sin\pars{\pi \over 2}\ln\pars{\Gamma\pars{1 \over 4}} + \sin\pars{\pi}\ln\pars{\Gamma\pars{1 \over 2}} + \sin\pars{3\pi \over 2}\Gamma\pars{3 \over 4}} \\[3mm]&=-\pi\bracks{\gamma + 2\ln\pars{2} + \ln\pars{2\pi}} +2\pi\bracks{\ln\pars{\Gamma\pars{1 \over 4}} - \ln\pars{\Gamma\pars{3 \over 4}}} \\[3mm]&=-\pi\bracks{\gamma + 2\ln\pars{2}} - 2\pi\bracks{% \ln\pars{\root{2\pi}} +\ln\pars{\Gamma\pars{3 \over 4} \over \Gamma\pars{1 \over 4}}} \\[3mm]&=-\pi\bracks{\gamma + 2\ln\pars{2}} -2\pi\ln\pars{\root{2\pi}\,{\Gamma\pars{3 \over 4} \over \Gamma\pars{1 \over 4}}} \end{align}
By replacing this result in $\pars{3}$, we find: $$\color{#00f}{\large% \int_{0}^{1}\ln\pars{\ln\pars{1 \over x}}\,{\dd x \over 1 + x^{2}} ={\pi \over 2}\,\ln\pars{\root{\vphantom{\large A}2\pi}\, {\Gamma\pars{3/4} \over \Gamma\pars{1/4}}}} $$
As an 'extra-bonus' we can use the identity $\ds{\Gamma\pars{z} = {\pi \over \Gamma\pars{1 - z}\sin\pars{\pi z}}}$ to 'kill' one of the $\Gamma\,$'s functions: $\ds{\Gamma\pars{1 \over 4} = {\root{2}\pi \over \Gamma\pars{3/4}}}$ which yields: $$ \int_{0}^{1}\ln\pars{\ln\pars{1 \over x}}\,{\dd x \over 1 + x^{2}} ={\pi \over 2}\,\ln\pars{\Gamma^{\,2}\pars{3/4} \over \root{\pi}} $$
See also V. Adamchik's formula $$\int_0^1 \frac{x^{p-1}}{1+x^n}\log \log \frac{1}{x}dx = \frac{\gamma+\log(2n)}{2n}(\psi(\frac{p}{2n})-\psi(\frac{n+p}{2n}))+\frac{1}{2n}(\zeta'(1,\frac{p}{2n})-\zeta'(1,\frac{n+p}{2n}))$$ in http://dx.doi.org/10.1145/258726.258736 .