I was observing this question by @Brightsun and conjecture $(1)$
$$\int_{0}^{\infty}e^{-\sqrt{x}}\ln{\left(1+{1\over \sqrt{x}}\right)}=2\gamma \tag1$$
An attempt
$x=u^2$ then $(1)$ becomes
$$2\int_{0}^{\infty}ue^{-u}\ln{\left(1+{1\over u}\right)}\tag2$$
$$2\sum_{n=0}^{\infty}{(-1)^n\over n!}\int_{0}^{\infty}u^{n+1}\ln{\left(1+{1\over u}\right)}\mathrm du\tag3$$
Changing $(2)$ by applying $e^x$, $(3)$ diverges.
How would one prove $(1)?$