I want to ask a quick question, i.e., how to calculate the below integral \begin{align*} I=\int_{0}^{\pi}dx\int_{0}^{\pi}dy \frac{1}{\sqrt{1+4\cos^2x+4\cos x\cos y}} \end{align*} I mean what's the closed-form of it.
I don't know if the substitution $u=\cos(x), v=\cos(y)$ is a good starting point. Thanks.
Here below is the code in ${\textit{Mathematica}}$
X = Integrate[(1/Sqrt[1 + 4*Cos[x]*Cos[x] + 4*Cos[x]*Cos[y]]) Boole[ 0 <= y < Pi && 0 <= x < Pi], {x, 0, Pi}, {y, 0, Pi}]; N[X, 5]
and the result is 8.8475.