Integrate $\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}dx$ 
Evaluate
$$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$$

I have tried substitution of $\sin x$ as well as $\cos x$ but it is not giving an answer.
Do not understand if there is a formula for this or not.

\begin{align}
I&=\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx\\
&=\int \frac{1}{\sqrt{\cot x}+1}dx
=\frac12x -\frac12 \int \frac{\sqrt{\cot x}-1}{\sqrt{\cot x}+1}dx\tag1 \\
\end{align}
where, with $t=\sqrt{\cot x}$
\begin{align}
\int \frac{\sqrt{\cot x}-1}{\sqrt{\cot x}+1}dx
&=\int \frac{2t(1-t)}{(1+t)(1 + t^4)}dt\\
&= 2 \int \frac{t^3}{1+t^4}-\frac1{1+t}+\frac{1-t^2}{1+t^4}\ dt\\
&= \frac12\ln(1+t^4)-2\ln(1+t)+2\int \frac{d(t+\frac1t)}{2-(t+\frac1t)^2}dt\\
&= 2\ln \frac{\sqrt[4]{1+t^4}}{1+t}+\sqrt2\coth^{-1}\frac{1+t^2}{\sqrt2t}
\end{align}
Substitute into (1) to obtain
\begin{align}
I= \frac12x+\ln (\sqrt{\sin x}+\sqrt{\cos x})
-\frac1{\sqrt2}\coth^{-1}\frac{\sqrt{\tan x}+\sqrt{\cot x}}{\sqrt{2}}
\end{align}
