Determine the arc length of the curve $y=\ln \left(\cos x\right)$ from $x=0$ to $x=\frac{\pi }{4}$.

$\ln (\sqrt{2}-1)$ units

$\ln (\sqrt{3}+1)$ units

$\ln (\sqrt{5}+1)$ units

$\ln (\sqrt{2}+1)$ units