If $I=\int \left(\sqrt{\cot x}-\sqrt{\tan x}\right)dx$, then $I=$

$I=\int \left(\frac{\sqrt{c}}{\sqrt{s}}-\frac{\sqrt{s}}{\sqrt{c}}\right)dx$

$=\int \frac{c-s}{\sqrt{\sin x\cos x}}dx$

$\sqrt{2}\int \frac{\cos x-\sin x}{\sqrt{\sin 2x}}dx$

$t=\sin x+\cos x$

$\Rightarrow t^2=1+\sin 2x$

$\sqrt{2}\int \frac{dt}{\sqrt{t^2-1}}$

$=\sqrt{2}\log \left|t+\sqrt{t^2-1}\right|$

$=\sqrt{2}\log \left|\sin x+\cos x+\sqrt{\sin 2x}\right|$

$=\sqrt{2}\log \left|\sin \left(x+\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)+\sqrt{\sin 2x}\right|$

$\Rightarrow \sqrt{2}\log \left|\sin x+\cos +\sqrt{\sin 2x}\right|$