Evaluate $\int_{-\infty}^{+\infty}\frac{1}{\sqrt{x^2+1}}\cdot\frac{1}{e^{\beta\sqrt{x^2+1}}+1}dx$ using contour integration with branch cuts

Question

I want to calculate the integral $$\int_{-\infty}^{+\infty}\frac{1}{\sqrt{x^2+1}}\cdot\frac{1}{e^{\beta\sqrt{x^2+1}}+1}dx$$ for $\beta>0$ with the residue theorem. I tried to follow the steps in this example, using the same contour, but I couldn't get any meaningful results.