I will work on it ...

I think the trick will be to multiply by [sqrt(x+1)-sqrt(x-1)]^2

Anyway, let's try and see how far we get

[sqrt(x+1)-sqrt(x-1)] * [sqrt(x+1)+sqrt(x-1)] = [sqrt(x+1)-sqrt(x-1)]^2 * (4x-1)/2

sqrt(x+1) * sqrt(x+1) + sqrt(x+1) * sqrt(x-1) - sqrt(x-1) * sqrt(x-1) - sqrt(x-1) * sqrt(x-1) = ...

(x+1) + (terms cancel each other) - (x-1) = ...

2 = [sqrt(x+1)-sqrt(x-1)] * [sqrt(x+1)-sqrt(x-1)] * (4x-1)/2

2 = [sqrt(x+1) * sqrt(x+1) - sqrt(x+1) * sqrt(x-1) - sqrt(x-1) * sqrt(x+1) + sqrt(x-1) * sqrt(x-1)] * (4x-1)/2

2 = [(x+1) - 2 * sqrt(x+1) * sqrt(x-1) + (x-1)] * (4x-1)/2

2 = [2x - 2 * sqrt(x+1) * sqrt(x-1) ] * (4x-1) / 2

4 = [2x - 2 * sqrt(x+1) * sqrt(x-1) ] * (4x-1)

Hmmm ... that helped a bit, let's try it again but using [sqrt(x+1)+sqrt(x-1)] * [sqrt(x+1)-sqrt(x-1)]