\[ \sqrt {x^2-1} y'(x)-\sqrt {y(x)^2-1}=0 \] ✓ Mathematica : cpu = 0.149295 (sec), leaf count = 173
\[\left \{\left \{y(x)\to -\frac {1}{2} e^{-c_1} \sqrt {2 x^2+2 e^{4 c_1} x^2-2 \sqrt {(x-1) (x+1)} x+2 e^{4 c_1} \sqrt {(x-1) (x+1)} x-1+2 e^{2 c_1}-e^{4 c_1}}\right \},\left \{y(x)\to \frac {1}{2} e^{-c_1} \sqrt {2 x^2+2 e^{4 c_1} x^2-2 \sqrt {(x-1) (x+1)} x+2 e^{4 c_1} \sqrt {(x-1) (x+1)} x-1+2 e^{2 c_1}-e^{4 c_1}}\right \}\right \}\] ✓ Maple : cpu = 0.006 (sec), leaf count = 29
\[\left \{c_{1}+\ln \left (x +\sqrt {x^{2}-1}\right )-\ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-1}\right ) = 0\right \}\]