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with

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

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

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

with

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

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

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

with

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

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

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

with

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

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

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

with

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

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

and $$d\tau=\sqrt{1-v^2}\,dt=X\,dt=\frac{1}{\alpha}\,dt$$

with

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

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

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