Finding the root of an equation involving digamma functions Is it possible to get an analytic solution for the equation
\begin{align} 
\frac{1}{x} + 2\psi(2x) + \pi \cot(\pi x) = 0
\end{align}
for $x\in(0,1)$ (using the Newton-Raphson method I get $x\approx 0.60778$)?.  If not, is it possible to get the solution within an interval? Using the reflection and duplication formula I get
\begin{align} 
\frac{1}{x} +2\log 2 + \psi(1-  x)  + \psi\left(x + \frac{1}{2}\right)  = 0.
\end{align}
Any suggestion or inequality to be used will be really appreciated.

I do not think that a closed form solution does exist. 
The solution $x=0.607778331746657190693779866290$ is not recognized by inverse symbolic calculators. It is "surprisingly" close to
$$\frac{3 I_0(1)+I_0(2)}{10} =0.607778294$$
