Is it possible to get symbolic integral for this?

Here is the function $$ I \;=\;\int_{\theta=0}^{\pi}\!\!\!\int_{\phi=0}^{2\pi} \dfrac{R^2\,\sin(\theta)\,\Bigl(\tfrac{a}{2} \;-\;R\,\sin(\theta)\,\cos(\phi)\Bigr)} {\bigl[a^2 \;-\; a\,R\,\bigl(\sin\theta\,\cos\phi \;+\;\sin\theta\,\sin\phi \;+\;\sqrt{2}\,\cos\theta\bigr)+R^2\bigr]^{3/2}} \;d\phi\,d\theta, $$

when $a>0,\quad R>0,\quad R
In my opinion, it should be able to get an symbolic answer without introducing any elliptic function.

However, this takes to long to evaluate in Mathematica and I finally to give up.

It should be $$\frac{2 \pi R^2} {a^2}$$

One can use a series expansion in $R$ to demonstrate the result for at least $\mathcal{O}(R^6)$.

Consider the following:

Therefore for the integrand we have:

And for its series expansion:

Now integrate:

Notice, no additional assumptions are needed, however, if a result for $R>a$ is desired, one has to expand over $R$ at infinity and follow the same procedure. It yields zero.

Let t == R/a

Numerically evaluate for various values of t

Find a fit to the data