with

$$t=X\,\sinh(T)\\
x=X\,\cosh(T)$$

hence
$$\frac{dx}{dt}=v=\frac{\sinh(T)}{\cosh(T)}$$


and
$$\gamma=\frac{1}{\sqrt{1-v^2}}=\frac{dt}{d\tau}\quad\Rightarrow\quad\\d\tau=\sqrt{1-v^2}\,dt=\sqrt {1-{\frac { \left( \sinh \left( T \right)  \right) ^{2}}{
 \left( \cosh \left( T \right)  \right) ^{2}}}}X\cosh \left( T
 \right)\,dT
=X\,dT=\frac{1}{\alpha}\,dT$$