I was wondering how one could break the integral
$$\int_{0}^{2\pi}\frac{1}{(a^2sin^2(\theta)+b^2cos^2(\theta))^{1/4}}d\theta$$
into other elliptic integrals of the first, second, and third kind. I came upon a thread that said all elliptic integrals can be broken down into elliptic integrals of the first, second, and third kind, and I noticed that another similar integral
$$\int_{0}^{2\pi}\frac{1}{(b^2sin^2(\theta)+a^2cos^2(\theta))^{1/4}}d\theta$$
was equal (or at least so I think). I feel like if I could break them down, equivalent elliptic factors could cancel out and I could prove their equivalence, but I don't know how to break them down. How do you break both integrals into their decompositions? Also, sorry this is my first question on Stack Exchange, so there are probably flaws everywhere in my question. Thanks!
To clarify, I'm asking if
$$\int_{0}^{2\pi}\frac{1}{(b^2sin^2(\theta)+a^2cos^2(\theta))^{1/4}}d\theta = \int_{0}^{2\pi}\frac{1}{(a^2sin^2(\theta)+b^2cos^2(\theta))^{1/4}}d\theta $$
is true or not for all REAL a and b.
Oh yeah, and here is a source I found that might clarify: https://pdfs.semanticscholar.org/3a41/3ed6779db174ff01c0debea7f41e12a215d0.pdf
And here is someone on Quora decomposing another elliptic integral: https://www.quora.com/How-do-I-integrate-displaystyle-int_0-frac-pi-2-frac-dx-sqrt-1+-sin-3x/answer/Luke-Gustafson