\input amstex \int \frac{dx}{x^2 \sqrt{x^2-1}} \overset{x=\sec \theta}\to= \int \frac{d\theta}{\sec \theta} = \int \cos \theta\, d\theta = \sin \theta + c =\sin(\operatorname{arcsec} x)+c= \sin\left(\arcsin\left(\frac{\sqrt{x^2-1}}{x}\right)\right) +c = \frac{\sqrt{x^2-1}}{x}+c