Find the arc length of the curve below on the given interval. $y=\ln \left(x-\sqrt{x^{2}-1}\right)$, for $1 \leq x \leq \sqrt{82}$ (Hint: Integrate with respect to $y$.) The length of the curve is (Simplify your answer.)

Step 1 ：Given the function \(y=\ln \left(x-\sqrt{x^{2}-1}\right)\), we need to find the arc length of the curve from \(x=1\) to \(x=\sqrt{82}\).

Step 2 ：The formula for the arc length of a curve defined by a function \(y=f(x)\) from \(x=a\) to \(x=b\) is given by \(L = \int_a^b \sqrt{1 + [f'(x)]^2} dx\).

Step 3 ：First, we need to find the derivative of the function. The derivative of \(y=\ln \left(x-\sqrt{x^{2}-1}\right)\) is \(y' = \frac{-x/\sqrt{x^{2}-1} + 1}{x - \sqrt{x^{2}-1}}\).

Step 4 ：Substitute the derivative into the formula, we get the integrand \(\sqrt{1 + \left(\frac{-x/\sqrt{x^{2}-1} + 1}{x - \sqrt{x^{2}-1}}\right)^2}\).

Step 5 ：The integral is quite complex and it's not easy to simplify it manually. However, we can simplify the integrand to \(\sqrt{\frac{x^2}{x^2 - 1}}\).

Step 6 ：Finally, we evaluate the integral from \(x=1\) to \(x=\sqrt{82}\) to get the arc length. The length of the curve is \(\boxed{9}\).