${\int }_{\frac{-1}{2}}^{\frac{1}{2}}\sqrt{\left\{\left.{\left[\frac{\left(x+1\right)}{x-1}\right]}^2+{\left(\frac{x-1}{x+1}\right)}^2-2\right\}\right.}dx=$

$I={\int }_{\frac{-1}{2}}^{\frac{1}{2}}\sqrt{{\left[\frac{x+1}{x-1}-\frac{x-1}{x+1}\right]}^2}dx={\int }_{\frac{-1}{2}}^{\frac{1}{2}}\left|\frac{4x}{x^2-1}\right|dx=-2{\int }_0^{\frac{1}{2}}\frac{4x}{x^2-1}dx$

$=-4{\left[\log \left|x^2-1\right|\right]}_0^{\frac{1}{2}}=-4\log \left(\frac{3}{4}\right)=4\log \left(\frac{4}{3}\right)=\log \left(\frac{256}{81}\right)$