I have the following inequality, which (supposedly) holds for every $x\in\mathbb{R}$:
$$ 1+x\ln\left(x+\sqrt{1+x^{2}}\right)\geq\sqrt{1+x^{2}} $$ I've been struggling to find some known inequalities involving logarithms that could be applied here (and lack of condition for $x$ to be non-negative doesn't help either). I would be very helpful for hints on how this should be approached.