\input amstex \int_1^{+\infty} \frac{\sqrt{x^2-1}}{x} \frac{1}{x^2+1}\, dx \overset{x=1/y}\to=  \int_1^0 \frac{\sqrt{1/y^2-1}}{1/y (1/y^2+1)}\, (-dy/y^2)
=\int_0^1 \frac{y \cdot y^2 \sqrt{(1-y^2)/y^2}}{(1+y^2) y^2}\, dy=\int_0^1 \frac{\sqrt{1-y^2}}{1+y^2}\, dy