Euclidean oscillator diagonal kernel

At imaginary duration $\beta \hslash$, $K_E(q,q)=\sqrt{m\omega /[2\pi \hslash \sinh x]}\exp [-m\omega q^2\tanh (x/2)/\hslash ]$, $x=\beta \hslash \omega$. Integrating the common endpoint yields $[2\sinh (x/2){]}^{-1}$.