Problem

The curve $C$ has equation

$$y = \frac{1}{7}\operatorname{arcosh}(2x) \qquad \frac{1}{2} \le x \le \frac{13}{2}.$$

Using calculus, determine the exact length of the curve $C$. Give your answer in the form $p\sqrt{q}$, where $p$ and $q$ are constants to be found.