Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $

$$y=\int\sqrt{\tan x}\,dx$$ $$g=\int\sqrt{\cot x}\,dx$$

\begin{align} y+g&=\int\left(\sqrt{\tan x}+\sqrt{\cot x}\right)\,dx \\&=\sqrt2\int\frac{\sin x+\cos x}{\sqrt{\sin2x}}\,dx \\& =\sqrt2\int\frac{(\sin x-\cos x)'}{\sqrt{1-(\sin x-\cos x)^2}}\,dx\\& =\sqrt2\int\frac{1}{\sqrt{1-u^2}}\,du \\& =\sqrt2\sin^{-1}u \\& =\sqrt2\sin^{-1}(\sin x-\cos x)\end{align}

\begin{align} y-g&=\int\left(\sqrt{\tan x}-\sqrt{\cot x}\right)\,dx \\& =\sqrt2\int\frac{\sin x-\cos x}{\sqrt{\sin2x}} \\& =-\sqrt2\int\frac{(\sin x+\cos x)'}{\sqrt{(\sin x+\cos x)^2-1}}\,dx \\& =-\sqrt2\int\frac{s'}{\sqrt{s^2-1}}\,ds \\& =-\sqrt2\cosh^{-1}(\sin x+\cos x) \end{align} \begin{align}y&=\frac{(y-g)+(y+g)}2 \\&= \frac{\sqrt2}2(\sin^{-1}(\sin x-\cos x)-\cosh^{-1}(\sin x+\cos x)) + C\end{align}

Let $I = \int\sqrt{\tan x}\;\mathrm{d}x$ and $J = \int\sqrt{\cot x}\;\mathrm{d}x$.

Now $$\begin{align}I + J &= \int\left(\sqrt{\tan x} + \sqrt{\cot x}\right) \;\mathrm{d}x \\ &= \sqrt{2} \int\frac{\sin x + \cos x}{\sqrt{\sin 2x}} \;\mathrm{d}x \\[5pt] &= \sqrt{2} \int\frac{(\sin x - \cos x)'}{\sqrt{1-(\sin x - \cos x)^2}} \;\mathrm{d}x \\[5pt] &= \sqrt{2} \sin^{-1}(\sin x - \cos x) + \mathbb{C_1} \tag{1} \\ \end{align}$$

and $$\begin{align}I - J &= \int\left(\sqrt{\tan x} - \sqrt{\cot x}\right) \;\mathrm{d}x \\ &= \sqrt{2} \int\frac{(\sin x - \cos x)}{\sqrt{\sin 2x}} \;\mathrm{d}x \\ &= -\sqrt{2} \int\frac{(\sin x + \cos x)'}{\sqrt{(\sin x + \cos x)^2 - 1}} \;\mathrm{d}x \\ &= -\sqrt{2} \ln\left|(\sin x + \cos x) + \sqrt{(\sin x + \cos x)^2 - 1}\right| + \mathbb{C_2} \tag{2} \\ \end{align}$$

Now, adding $(1)$ and $(2)$:

$$I = \frac{1}{\sqrt{2}} \sin^{-1}(\sin x - \cos x) - \frac{1}{\sqrt{2}} \ln\left|\sin x + \cos x + \sqrt{\sin 2x} \vphantom{x^{x^x}} \right| + \mathbb{C}$$

Let $u = \sqrt{\tan x}$, then $u^2 = \tan x$. Thus $2u\;\mathrm{d}u = \sec^2 x\;\mathrm{d}x = (u^4 + 1)\mathrm{d}x$. Thus $\mathrm{d}x = \dfrac{2u\;\mathrm{d}u}{u^4 + 1}$. So:

$$\int\sqrt{\tan x}\;\mathrm{d}x = \int\frac{2u^2}{u^4+1}\;\mathrm{d}u$$ You can take it from here.