1. **State the problem:** Simplify the expression $$\left[\sqrt{s-1} + \ln \sqrt{s^{2} - 1^{2}}\right]^{-1}$$. 2. **Rewrite the expression inside the logarithm:** Note that $$1^{2} = 1$$, so $$s^{2} - 1^{2} = s^{2} - 1$$. 3. **Simplify the logarithm:** Using the property $$\ln \sqrt{x} = \frac{1}{2} \ln x$$, we have $$\ln \sqrt{s^{2} - 1} = \frac{1}{2} \ln (s^{2} - 1).$$ 4. **Rewrite the entire expression inside the brackets:** $$\sqrt{s-1} + \frac{1}{2} \ln (s^{2} - 1).$$ 5. **Final expression:** The original expression is the inverse of this sum, so $$\left[\sqrt{s-1} + \frac{1}{2} \ln (s^{2} - 1)\right]^{-1} = \frac{1}{\sqrt{s-1} + \frac{1}{2} \ln (s^{2} - 1)}.$$