= Solution
A single formula valid on both real components is
$$
\boxed{R(x)=\ln\left|x+\sqrt{x^2-1}\right|,\qquad |x|>1.}
$$
Let $s=\sqrt{x^2-1}>0$. Since $x+s\ne0$,
$$
R'(x)=\frac{1+x/s}{x+s}=\frac1s=\frac1{\sqrt{x^2-1}}.
$$
Equivalently, $R(x)=\operatorname{arcosh}x$ for $x>1$ and $R(x)=-\operatorname{arcosh}(-x)$ for $x<-1$. The graph is odd for this choice of constants, increases on both components, and approaches zero with vertical tangent as $x\to1^+$ or $x\to-1^-$. Its far-field behaviors are $R(x)\sim\ln(2x)$ on the right and $R(x)\sim-\ln(-2x)$ on the left. The interval $[-1,1]$ is excluded. Independent additive constants could again be assigned to the two components. The right panel sketches these <real inverse-hyperbolic antiderivative branches>.
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