Mathematical formulae for Complex Arcus Cotangents?

I'm writing reusable complex functions for use in Shaders.

I've done most of the complex series. All sines, cosines, and tangents (arcus, hyperbolic, and hyperbolic arcus variations as well)

However, I'm missing the last 2 arcus cotangents. I couldn't find any documentation on the function's definitions.

All I have is the complex cotangent and hyperbolic definitions, which are both pretty simple and obvious.

The Complex Arcus functions dont follow the same rules, so I dont know the definitions.

(All c Functions are complex functions)

$$Complex\space Cotangent\left(z\right)=cDiv\left(cCos\left(z\right),\ cSin\left(z\right)\right)$$

$$Complex\space Hyperbolic\space Cotangent\left(z\right)=cDiv\left(cCosH\left(z\right),\ cSinH\left(z\right)\right)$$

For example, here's the definition of the Complex Arcus Tangent: (Complex log uses base e)

$$Complex\space Arcus\space Tangent\left(z\right)=cDiv\left(cLog\left(cMul\left(z,\ 1i\right)+1\right)-cLog\left(cMul\left(z,\ 1i\right)-1\right),\ 2i\right)$$

It's very different from the typical sin / cos = tan rule

TL;DR: Im trying to find both of the Complex Arcus Cotangent and Complex Hyperbolic Arcus Cotangent