Exact expression of a trigonometric integral Let $a>2$ be a real number and consider the following integral
$$
I(a)=\int_0^\pi\int_0^\pi \frac{\sin^2(x)\sin^2(y)}{a+\cos(x)+\cos(y)} \mathrm{d}x\,\mathrm{d}y
$$

My question. Does there exist a closed-form expression of $I(a)$?

Some comments. Since $a-2<a+\cos(x)+\cos(y)<a+2$ and $\int_0^\pi \int_0^\pi \sin^2(x)\sin^2(y)\ \mathrm{d}x\, \mathrm{d}y=\frac{\pi^2}{4}$, we have the following bounds
$$
\frac{\pi^2}{4(a+2)} < I(a) < \frac{\pi^2}{4(a-2)},
$$ 
however I didn't manage to find an exact expression for $I(a)$. Any help is welcome!

With CAS help:
$$\int _0^{\pi }\int _0^{\pi }\frac{\sin ^2(x) \sin ^2(y)}{a+\cos (x)+\cos (y)}dydx=\\\mathcal{L}_q\left[\int _0^{\pi }\int _0^{\pi }\mathcal{L}_a^{-1}\left[\frac{\sin ^2(x) \sin ^2(y)}{a+\cos (x)+\cos
   (y)}\right](q)dydx\right](a)=\\\mathcal{L}_q\left[\int_0^{\pi } \left(\int_0^{\pi } e^{-q (\cos (x)+\cos (y))} \sin ^2(x) \sin ^2(y) \, dx\right) \, dy\right](a)=\\\mathcal{L}_q\left[\int_0^{\pi }
   \frac{e^{-q \cos (y)} \pi  I_1(q) \sin ^2(y)}{q} \, dy\right](a)=\\\mathcal{L}_q\left[\frac{\pi ^2 I_1(q){}^2}{q^2}\right](a)=\\\frac{a \pi ^2}{2}-\frac{2}{3} a \pi  E\left(\frac{4}{a^2}\right)-\frac{1}{6}
   a^3 \pi  E\left(\frac{4}{a^2}\right)-\frac{2}{3} a \pi  K\left(\frac{4}{a^2}\right)+\frac{1}{6} a^3 \pi  K\left(\frac{4}{a^2}\right)=\\\frac{\pi ^2 \, _3F_2\left(\frac{1}{2},1,\frac{3}{2};2,3;\frac{4}{a^2}\right)}{4 a}$$
for: $a>2$
$$\frac{1}{6} \pi  \left(-2 \left(a^2-4\right) K\left(\frac{a^2}{4}\right)-2 \left(a^2+4\right) E\left(\frac{a^2}{4}\right)+3 \pi  a\right)$$
for: $a<2$
where: $K$,$E$ gives the elliptic integral of the first kind  and second kind.
Mathematica code:
HoldForm[Integrate[(Sin[x]^2*Sin[y]^2)/(a + Cos[x] + Cos[y]), {x, 0, Pi}, {y, 0, Pi}] == (a \[Pi]^2)/2 - 2/3 a \[Pi] EllipticE[4/a^2] - 1/6 a^3 \[Pi] EllipticE[4/a^2] - 2/3 a \[Pi] EllipticK[4/a^2] + 1/6 a^3 \[Pi] EllipticK[4/a^2] == Pi^2/(4 a)*HypergeometricPFQ[{1/2, 1, 3/2}, {2, 3}, 4/a^2]] // TeXForm
Plot a solution:
f[a_?NumericQ] := NIntegrate[(Sin[x]^2*Sin[y]^2)/(a + Cos[x] + Cos[y]), {x, 0, Pi}, {y,0, Pi}]; 
g[a_] := (a \[Pi]^2)/2 - 2/3 a \[Pi] EllipticE[4/a^2] - 
1/6 a^3 \[Pi] EllipticE[4/a^2] - 2/3 a \[Pi] EllipticK[4/a^2] + 
1/6 a^3 \[Pi] EllipticK[4/a^2]; Plot[{f[a], g[a]}, {a, 2, 20}, 
PlotStyle -> {Red, {Dashed, Black}}, PlotLabels -> {"integral", "Analytic solution"}]


