Using formula 288.03 in Byrd and Friedman's handbook:
$\int_{\varphi}^{\pi/2}\frac{\sin^m u}{\sqrt{a+b\sin\;u}}\mathrm du=\frac2{\sqrt{a+b}}\int_0^{F\left(\arcsin\left(\sqrt{\frac{1-\sin\;\varphi}{2}}\right)\mid\frac{2b}{a+b}\right)}\left(1-2\mathrm{sn}^2\left(u\mid\frac{2b}{a+b}\right)\right)^m\mathrm du$
for $a > b > 0$, where $F(\phi|m)$ is the incomplete elliptic integral of the first kind and $\mathrm{sn}(u|m)$ is a Jacobian elliptic function, we have
$\int_{-\pi/2}^{\pi/2} \frac{\sin y}{\sqrt{\sin y+c}}\mathrm{d}y=2\left(\sqrt{1+c}\;E\left(\frac2{1+c}\right)-\frac{c}{\sqrt{1+c}}K\left(\frac2{1+c}\right)\right)$
where $K(m)$ is the complete elliptic integral of the first kind, and $E(m)$ is the complete elliptic integral of the second kind.