2.1.10.6 ✓ Mathematica. Time used: 0.012 (sec). Leaf size: 18

ode=D[x[t],t]==Sqrt[x[t]^2-1]; 
ic={x[0]==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} \left (e^{-t}+e^t\right ) \end{align*}