Reference for classical action of the standard height function on the two-torus written as sum of elementary and elliptic integrals?

The height function of the two torus $\mathbb{T}^2$ is a standard example. It has $2$ hyperbolic points and $2$ elliptic points. I was wondering if there exists a reference that computes the classical action of this function over the set of regular values?

Obtaining the actual formulas seems doable, but maybe a bit messy since we will be working with two angles. So therefore I was also wondering if there is a reference where this is actually written as a sum of elementary integrals and elliptic integrals?

Thanks in advance.