= Real inverse-hyperbolic antiderivative branches
{title2=$\int dx/\sqrt{x^2-a^2}=\ln|(x+\sqrt{x^2-a^2})/a|+C$}
For $a>0$ and $|x|>a$, the displayed real <antiderivative> equals $\operatorname{arcosh}(x/a)$ on the right component and $-\operatorname{arcosh}(-x/a)$ 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.
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