Evaluate: `int_(-1/2)^(1/2)[((x+1)/(x-1))^2+((x-1)/(x+1))^2-2]^(1/2)dx`

Explanation

Given integral
`int_(-1//2)^(1//2)[{(x+1)/(x-1)-(x-1)/(x+1)}^(2)]^(1//2)dx`
`=int_(-1//2)^(1//2)|(x+1)/(x-1)-(x-1)/(x+1)|dx`
`= int_(-1//2)^(1//2)|(4x)/(x^(2)-1)|dx=2int_(0)^(1//2)|(4x)/(x^(2)-1)|dx`
`=2int_(0)^(1//2)(4x)/(1-x^(2))dx`
`[ :' |(4x)/(x^(2)-1)|=-(4x)/(x^(2)-1)` when `0lexle1/2]`
`=-4[log (1-x^(2))]_(0)^(1//2)`
`=-4log(3//4)`
`=4log(4//3)`