There are several results in Spiegel's "Mathematical Handbook of Formulas and Tables" (Schaum, 1968) concerning $\cosh^{-1}$ which are presented in the following format:
$$\cosh^{-1} x = \pm \left\{{\ln (2x) - \left({\dfrac 1 {2 \cdot 2x^2} + \dfrac {1 \cdot 3} {2 \cdot 4 \cdot 4x^4} + \dfrac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 \cdot 6x^6} }\right) }\right\}$$
$$[+ \text { if }\cosh^{-1} x > 0, x \ge 1, - \text { if }\cosh^{-1} x < 0, x \ge 1]$$
(item $20.40$)
and:
$$\int \cosh^{-1} \frac x a \, \mathrm d x = \begin{cases} x \cosh^{-1} (x/a) - \sqrt {x^2 - a^2} & : \cosh^{-1} (x/a) > 0 \\ x \cosh^{-1} (x/a) + \sqrt {x^2 - a^2} & : \cosh^{-1} (x/a) < 0 \end{cases}$$
(item $14.651$)
There are a number of such.
What confuses me is that I have been led to believe that the (real) inverse hyperbolic cosine is specifically defined as being the positive value of $y$ such that $\cosh y = x$.
So what is the motivation and conventional interpretation of "$f (\cosh^{-1} x)$ such that $\cosh^{-1} x < 0$"?
The 1978 edition of the Mathematical Handbook says that "$\text{Arg ch}(x)=\ln(x+\sqrt{x^2-1}),x\ge0$, ($\text{Arg ch}(x)>0$ is the principal branch)" (translated from French). This fully justifies the formulas that you mention, which pertain to the two possible branches (the two solutions in $y$ of $x=\cosh(y)$).
Technically speaking, the hyperbolic cosine is not an invertible function, and the notation $\cosh^{-1}(x)$ is no so valid. The function $\text{arcosh}(x)$ is well-known and "by default" taken to be the principal branch.