There is a curious four parameter iteration introduced by Borchardt:
$$a_{n+1}=\frac{a_n+b_n+c_n+d_n}{4} \\ b_{n+1}=\frac{\sqrt{a_n b_n}+\sqrt{c_n d_n}}{2} \\ c_{n+1}=\frac{\sqrt{a_n c_n}+\sqrt{b_n d_n}}{2} \\ d_{n+1}=\frac{\sqrt{a_n d_n}+\sqrt{b_n c_n}}{2} $$
Apparently, the limit of this iteration, denoted hereafter $B(a_0,b_0,c_0,d_0)$ has a closed form in terms of a certain double integral. However, this 'closed form' doesn't check out when I try implementing it in Mathematica.
For a complete description see this paper, section 2.5. For more general information see this paper.
Here is the description of the closed form from the first link above. First, define:
$$A = a_0 + b_0 + c_0 + d_0 \\ B = a_0 + b_0 - c_0 - d_0 \\ C = a_0 - b_0 + c_0 - d_0 \\ D = a_0 - b_0 - c_0 + d_0$$
(Note the problem - some of the numbers above can be negative).
Then define:
$$B_1=\frac{\sqrt{A B}+\sqrt{C D}}{2} \\ B_2=\frac{\sqrt{A B}-\sqrt{C D}}{2} \\ C_1=\frac{\sqrt{A C}+\sqrt{B D}}{2} \\ C_2=\frac{\sqrt{A C}-\sqrt{B D}}{2} \\ D_1=\frac{\sqrt{A D}+\sqrt{B C}}{2} \\ D_2=\frac{\sqrt{A D}-\sqrt{B C}}{2} $$
(Note the problem - some of the numbers above can be complex).
Then define:
$$\Delta=\sqrt[4]{ABCDB_1C_1D_1B_2C_2D_2}$$
And finally, the main parameters:
$$\alpha_0=\frac{A C B_1}{\Delta} \\ \alpha_1=\frac{C C_1 D_1}{\Delta} \\ \alpha_2=\frac{A C_2 D_1}{\Delta} \\ \alpha_3=\frac{B_1 C_1 C_2}{\Delta}$$
And define a function:
$$R(x):=x(x-\alpha_0)(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)$$
According to the paper, the limit of the iteration above is equal to:
$$\frac{\pi^2}{B(a_0,b_0,c_0,d_0)}=\int _0^{\alpha_3}\int _{\alpha_2}^{\alpha_1}\frac{x-y}{\sqrt{R(x) R(y)}}dxdy \tag{1}$$
I tried to implement this 'closed form' in Mathematica. However, for most initial conditions, even simple ones, Mathematica has trouble computing the integral and gives complex values anyway.
How can I tell if $(1)$ is correct? Is there a typo somewhere in my formulas? Or maybe there is some typo in the linked paper?
I want to know if $(1)$ is correct, and if not what is the correct form. I don't need the full proof. I tried to find Borchardt's original paper, but I couldn't. And it's not in English anyway.