How to derive elliptic integral of the first kind from $\int_{-\pi}^{\pi}\frac{1}{\sqrt{(\eta^{2}/2 - 3/2 -\cos 2\theta)^2 -4 \cos^2\theta}}\ d\theta$

Question

Spent several days already trying to figure out how to convert the given integral to this: $$\frac{8}{\sqrt{(\eta-1)^3(\eta+3)}}K\left(\sqrt{\frac{16\eta}{(\eta-1)^3(\eta+3)}}\right)$$ where $$ K\left(\sqrt{\frac{16\eta}{(\eta-1)^3(\eta+3)}}\right) = \int_{0}^{1}\left((1-x^2)\left(1-\sqrt{\frac{16\eta}{(\eta-1)^3(\eta+3)}} x^2\right)\right)^{-1/2}\ dx $$ This is a step from the problem of graphene density of states following the next article https://arxiv.org/abs/1705.08120 Please, help >_<

Answer 1

Hint

Let $$a=\frac{\eta^2-3}2 \qquad \text{and} \qquad\theta=\tan^{-1}(x)$$ to obtain $$I=\frac 1{a+1 }\int \frac {dx} {\sqrt{x^4+b x^2+c}}$$ where $$b=\frac{2 a^2-1}{(a+1)^2}\qquad \text{and} \qquad c=\frac{a^2-2 a+2}{(a+1)^2}$$

Let $r$ and $s$ be the roots of the quadratic in $x^2$ $$\int\frac{dx}{\sqrt{\left(x^2-r\right) \left(x^2-s\right)}}=\frac 1 {\sqrt s}F\left(\sin ^{-1}\left(\frac{x}{\sqrt{r}}\right)|\frac{r}{s}\right)$$