 Real inverse-hyperbolic antiderivative branches

For $a>0$ and $|x|>a$, the displayed real antiderivative equals $\mathrm{arcosh}(x/a)$ on the right component and $-\mathrm{arcosh}(-x/a)$ on the left, up to constants. Differentiating the absolute logarithm gives $1/\sqrt{x^2-a^2}$ on both components. The constants are independent because the domain is disconnected. The integrals from either finite singular boundary are improper integrals that converge, since their integrands diverge only like the inverse square root of distance to that boundary.