The integral `int (sqrt(x+1)-sqrt(x-1))/sqrt(x^2-1) dx` has to be determined.

`int (sqrt(x+1)-sqrt(x-1))/sqrt(x^2-1) dx`

= `int (sqrt(x+1)-sqrt(x-1))/sqrt((x - 1)(x +1)) dx`

= `int (sqrt(x+1)-sqrt(x-1))/(sqrt(x - 1)*sqrt(x +1)) dx`

= `int sqrt(x+1)/(sqrt(x-1)*sqrt(x+1)) - sqrt(x-1)/(sqrt(x+1)*sqrt(x-1)) dx `

= `int 1/sqrt(x-1) - 1/sqrt(x+1) dx` ` `

= `sqrt(x - 1)/(1/2) - sqrt(x+1)/(1/2)`

= `2*sqrt(x - 1) - 2*sqrt(x+1)`

The integral `int (sqrt(x+1)-sqrt(x-1))/sqrt(x^2-1) dx = 2*sqrt(x - 1) - 2*sqrt(x+1) + C`