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.

Recall that $$\begin{align}\sin(x) &= \frac{\tan(x)}{\sec(x)} \\ \cos(x) &= \frac1{\sec(x)} \\ \sec^2(x) &= 1 + \tan^2(x)\end{align}$$
Then,
$$\int\frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)} + \sqrt{\cos(x)}}\,\mathrm dx\equiv\int\sec^2(x)\cdot\frac{\sqrt{\tan(x)}}{\sqrt{\tan(x)} \left(1 + \tan^2(x)\right) + 1 + \tan^2(x)}\,\mathrm dx$$
Now, let $u = \sqrt{\tan(x)}\implies2\cdot\mathrm du = \dfrac{\sec^2(x)}{\sqrt{\tan(x)}}\,\mathrm dx$. So,
$$\begin{align}\int\sec^2(x)\cdot\frac{\sqrt{\tan(x)}}{\sqrt{\tan(x)} \left(1 + \tan^2(x)\right) + 1 + \tan^2(x)}\,\mathrm dx&\equiv\int2\cdot\frac{u^2}{u\left(1 + u^4\right) + 1 + u^4}\,\mathrm du \\ &= 2\int\frac{u^2}{(1 + u)(1 + u^4)}\,\mathrm du\end{align}$$
From here on, partial fractions and linearity are your friends.
