Definition:Real Inverse Hyperbolic Cotangent/Also known as

Real Inverse Hyperbolic Cotangent: Also known as
The real inverse hyperbolic cotangent is also known as the (real) area hyperbolic cotangent, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Note that as the real hyperbolic cotangent $\coth$ is injective, its inverse is properly a function on its domain.

Hence there is no need to make a separate distinction between branches in the same way as for real inverse hyperbolic cosine and real inverse hyperbolic secant.

Some sources refer to it as hyperbolic arccotangent, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cotangent.