Fourier transform $\left[\mathrm{csch}(x+i\epsilon-t)\right]^n\left[\mathrm{csch}(x+i\epsilon+t)\right]^m$

Question

In a physics related problem, I am trying to compute the Fourier transform \begin{align} \mathcal{F}\left[\frac{1}{\sinh^{n}\left[\pi T_R\left(x+ i\epsilon-t\right)\right]\sinh^{m}\left[\pi T_L\left(x+i\epsilon+t\right)\right]}\right]\left(\omega\right)=\int_{-\infty}^\infty\frac{dt e^{i\omega t}}{\sinh^{n}\left[\pi T_R\left(x+ i\epsilon-t\right)\right]\sinh^{m}\left[\pi T_L\left(x- i\epsilon+t\right)\right]}, \end{align} where $T_{L,R}$ are constants and $n,m$ are natural numbers. Employing the convolution theorem, we may split the Fourier transform as \begin{align} \mathcal{F}\left[\frac{1}{\sinh^{n}\left[\pi T_R\left(x+ i\epsilon-t\right)\right]\sinh^{m}\left[\pi T_L\left(x- i\epsilon+t\right)\right]}\right]\left(\omega\right)=\mathcal{F}\left[\frac{1}{\sinh^{n}\left[\pi T_R\left(x+ i\epsilon-t\right)\right]}\right]\left(\omega-\omega'\right)\ast\mathcal{F}\left[\frac{1}{\sinh^{m}\left[\pi T_L\left(x- i\epsilon+t\right)\right]}\right]\left(\omega'\right), \end{align} where \begin{align} \left(f*g\right)\left(\omega\right)=\int_{-\infty}^\infty d\omega' f\left(\omega-\omega'\right)g\left(\omega'\right). \end{align} Therefore, we have to compute the Fourier transforms of each one separately. This was my idea, but it seems impossible to compute the total convolution since the Fourier transform of $\left(\mathrm{csch}x\right)^n$ is a Euler Beta function. Any ideas (maybe different from this one) on how to proceed?