Express definite integral in terms of special functions?

Question

The integral is $$\int_0^{2\pi} \sqrt{1+a \sin x}\, dx$$ where $a$ is a parameter with $|a|<1$. I don't believe this can be expressed in terms of elementary functions, but surely it is related to some kind of special function? My main concern is a concise way to reference the function in a paper, I can use numerical evaluation to get the properties I need. My apologies if this has come up before, it is hard to search on.

Answer 1

As noted in comments, the closest known special function is $$\mathrm{E}(k)=\int_0^{\pi/2}\sqrt{1-k^2\sin^2 t}\,dt,$$ the complete elliptic integral of the second kind.

The given integral equals $I(a)=2\int_0^\pi\sqrt{1+a\cos y}\,dy$ (using $x=y+\pi/2$ and periodicity). Since clearly $I(a)=I(-a)$, we may assume $0\leqslant a\leqslant 1$. Then, substituting $y=2t$, we get $$I(a)=4\int_0^{\pi/2}\sqrt{1+a(1-2\sin^2 t)}\,dt=4\sqrt{1+a}\,\mathrm{E}\left(\sqrt{\frac{2a}{1+a}}\right).$$