special relativity - Rindler motion constant acceleration proper time

I have problem to calculate the proper time for Rindler coordinates: the coordinates in Minkowski space with constant acceleration are given by:

$$\begin{alignat}{7} t' &~=~ \frac{c}{g} \, && \sinh{\left(\frac{g}{c\tau}\right)} \\ x' &~=~ \frac{c^2}{g} \, && \cosh{\left(\frac{g}{c\tau}\right)} \end{alignat}$$

The proper time is

$$\tau_{\text{p}}~=~\int{\sqrt{1 - \frac{v^2 \left(t' \right)}{c^2}}}\, \mathrm{d}t' \,,$$ with $$\begin{alignat}{7} t' & ~=~ \frac{c}{g} \, && \sinh{\left(\frac{g}{c\tau}\right)} \\ \mathrm{d}t' & ~=~ && \cosh{\left(\frac{g}{c\tau}\right)} \,. \end{alignat}$$

How can I calculate $v \left(t' \right)$?