<s>[INST] Indefinite Integral of $\sqrt{\sin x}$ $$\int \sqrt{\sin x} ~dx.$$
Does there exist a simple antiderivative of $\sqrt{\sin x}$? How do I integrate it? 
 [/INST] Since $\sqrt{\sin(x)} = \sqrt{1 - 2 \sin^2\left(\frac{\pi}{4} -\frac{x}{2}\right)}$, this matches with the elliptic integral of the second kind:
$$\begin{align*}
   \int \sqrt{\sin(x)} \, \mathrm{d} x &\stackrel{u = \frac{\pi}{4}-\frac{x}{2}}{=} -2 \int \sqrt{1-2 \sin^2(u)} \,\mathrm{d} u\\ &= -2 E\left(u\mid 2\right) + c =  -2 E\left(\frac{\pi}{4}-\frac{x}{2}\middle|\, 2\right) + c
\end{align*}$$
where $c$ is an integration constant.
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