Do you have any idea to present integral form of this function?
$f(x)=\frac{1}{x^2}+\frac{1}{x}(\psi(x)-2\ln x-2)+2(1+\ln x)\psi^{'}(x)+x\ln x\psi^{''}(x).$ Where $\psi^{(n)}(x)$ is polygamma function and $x>0.$
Note: The function $f$ is Completely Monotonic function in $(0,\infty).$ Then, by Bernstein's theorem we obtain $f(x) = \int_0^\infty e^{-tx} \,dg(t).$
https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions