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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.

Cass Sackett
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  • You can ask WolframAlpha such questions (https://www.wolframalpha.com/input?i=integral+of+sqrt%281+%2B+a+sin+x%29+from+x+%3D+0+to+2pi); in this case I get "standard computation time exceeded"... Another option is to compute the first few terms of the Taylor series and search them in the OEIS (you may have to search the numerators or multiply by factorials): https://oeis.org/ – Qiaochu Yuan Sep 26 '22 at 17:01
  • I'll take a look using mathematica when i get the chance ig – Captain Chicky Sep 26 '22 at 17:45
  • This seems like an integral that can be written in terms of the elliptic integrals of the second kind. – HackR Sep 26 '22 at 17:50
  • Yes! It takes a little manipulation so it's not immediately obvious, but you can put it in terms of the elliptic integral. – Cass Sackett Sep 26 '22 at 18:05
  • done ConditionalExpression[ 2 (Sqrt[1 - a] EllipticE[(2 a)/(-1 + a)] + Sqrt[1 + a] EllipticE[(2 a)/(1 + a)]), Im[a] != 0 || -1 <= Re[a] <= 1] – Captain Chicky Sep 26 '22 at 23:36

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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 $$I(a)=4\int_0^{\pi/2}\sqrt{1+a(1-2\sin^2 t)}\,dt=4\sqrt{1+a}\,\mathrm{E}\left(\sqrt{\frac{2a}{1+a}}\right).$$

metamorphy
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