El valor de la integral $I = \int (\sqrt{\tan x} + \sqrt{\cot x}) dx$, donde $x \in \left( 0, \frac{\pi}{2} \right) \cup \left( \pi, \frac{3\pi}{2} \right)$, es:

(a) $\sqrt{2} \tan^{-1}\left( \frac{\sqrt{\tan x} - \sqrt{\cot x}}{\sqrt{2}} \right) + c$
(b) $\sqrt{2} \tan^{-1}\left( \frac{\sqrt{\tan x} + \sqrt{\cot x}}{\sqrt{2}} \right) + c$
(c) $-\sqrt{2} \tan^{-1}\left( \frac{\sqrt{\tan x} - \sqrt{\cot x}}{\sqrt{2}} \right) + c$
(d) $-\sqrt{2} \tan^{-1}\left( \frac{\sqrt{\tan x} + \sqrt{\cot x}}{\sqrt{2}} \right) + c$