How to implement this operator in Mathematica?

I am interested to implement this operator:

$$(ln D)[f] =left(frac{i pi}{2}-gammaright) f(x)+frac{1}{2},text{P.V.}int_{-infty}^infty left(frac{1}{x-x’}-frac{1}{| x-x’| }right),f(x’),dx’$$

The second term is problematic. Any suggestions?

P.S.
Some expected outputs:

$begin{align*}
(ln D) 1 & {}= -ln x -gamma
(ln D) x^n & {}= x^n (psi (n+1)-ln x)
(ln D) ln x & {}= -zeta(2) -(gamma+ln x)ln x.
end{align*}$