\text{Integrate}\left[\frac{\text{Coth}\left[\frac{6 *J *S \sqrt{1-\frac{1}{9} (\text{Cos}[a*x]+\text{Cos}[a*y]+\text{Cos}[a*z])^2}}{T}\right]}{\sqrt{1-\frac{1}{9}*(\text{Cos}[a*x]+\text{Cos}[a*y]+\text{Cos}[a*z])^2}},\left\{x,0,\frac{2*\pi }{a}\right\},\left\{y,0,\frac{2*\pi }{a}\right\},\left\{z,0,\frac{2*\pi }{a}\right\},\text{Assumptions}\to \{a>0\&\&a\in \text{Reals},S>0\&\&S\in \text{Reals},J>0\&\&J\in \text{Reals},T>0\&\&T\in \text{Reals}\}\right]