Why MMA refuses to simplify the following simple expression?

I have the following simple expression

Exp((Log(1 + x)^2 - Log(x)^2)/Log(1 + 1/x)) - x^2

that can be shown to simplify to x for positive x (and same for $x < -1$ if principal branch of log is taken, $log (x) = i pi + log (-x)$). However, none of the following

Simplify(Exp((Log(1 + x)^2 - Log(x)^2)/Log(1 + 1/x)) - x^2)
FullSimplify(Exp((Log(1 + x)^2 - Log(x)^2)/Log(1 + 1/x)) - x^2)
Simplify(Exp((Log(1 + x)^2 - Log(x)^2)/Log(1 + 1/x)) - x^2, 
 Assumptions -> {x > 0})
FullSimplify(Exp((Log(1 + x)^2 - Log(x)^2)/Log(1 + 1/x)) - x^2, 
 Assumptions -> {x > 0})

finds this simplification. Why is that? To show that the expression simplifies accordingly, use $a^2 – b^2 = (a+b)(a-b)$ formula inside the exponential and in the denominator $log (1+1/x) = log (1+x) – log(x)$. Some stuff cancels out and you’ll be left with $exp log (x+x^2) – x^2 = x$. I have no clue why MMA didn’t choose to go down this route, I assumed that the common formulas like $a^n – b^n$ are known to it. I tried Simplify(Log(1 + 1/x) + Log(x), Assumptions -> {x > 0}) and it correctly returned log(1+x), so this might not be the issue here.

This specific expression came up in some wider context, I frequently use MMA to simplify expressions after integration to get some nicer form and was surprised that this couldn’t be simplified.