Finding the root of an equation involving digamma functions

Question

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.

Answer 1

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 $$\frac{3 I_0(1)+I_0(2)}{10} =0.607778294$$