How to rewrite $f\left(x\right)=\frac{A^{2}e^{\frac{1}{2}x}+4e^{-\frac{1}{2}x}}{2A}$ in terms of hyperbolic functions

Question

I was solving for the function whose surface of revolution equals the volume of the solid of revolution. I ended up with two functions $f(x)=0$ and $$f\left(x\right)=\frac{A^{2}e^{\frac{1}{2}x}+4e^{-\frac{1}{2}x}}{2A}$$ then I used desmos to evaluate the integrals of the volume and surface area of the function and they were equal. The function looks similar to the $\cosh$ function but I'm not sure how to write it in terms of a hyperbolic function.