I am wondering whether there is a closed form for the following integral for $n\in\mathbb{N}$:

$$\gamma(n)=\int_0^1...\int_0^1\frac{1}{\left(1+\sqrt{1+x_1^2+...+x_n^2}\right)^{n+1}}\;dx_1...dx_n\tag{*}$$

Particular values which I am aware of include:

$$\gamma(1)=\frac{4\sqrt{2}-5}{3}$$

$$\gamma(2)=\frac{5}{4}-\frac{9\sqrt{3}}{8}+\frac{\pi}{4}$$

Both of these values were obtained by evaluating different integrals to the above which solved the same problem (see below), and I am not sure how to attack $(*)$ directly; I can see no good way of solving this integral. I am wondering especially about the value of $\gamma(3)$; does it also have a simple closed form? Is there a closed form for all $n$?

**Background**: I posted a question here about calculating the proportion $p(n)$ of an $n$-cube closer to the centre than to the outside, which seems to me like an interesting problem. The $n=2$ case is simple to solve in terms of the following integral:

$$p(2)=2\int_{0}^{\sqrt{2}-1}\frac{1-x^2}{2}-x\;dx=\frac{4\sqrt{2}-5}{3}$$

I was able to write $p(3)$ as follows:

$$p(3)=6\int_{0}^{\frac{\sqrt{3}-1}{2}}\int_{z}^{\sqrt{2-z^2}-1}\frac{1-x^2-z^2}{2}-x\;dx\;dz$$

and I managed to evaluate this to $\frac{5}{4}-\frac{9\sqrt{3}}{8}+\frac{\pi}{4}$, but I was not able to use my method to solve the problem for higher dimensions. In the comments, however, achille hui made a proposition that we have $p(n)=\gamma(n-1)$ for all $n$ and although I still do not perfectly understand his reasoning, the claim does check out numerically for the two values I know already. Furthermore, the new integral is in a nice simple symmetric form (unlike the methods I had been using which required a case-by-case analysis for every dimension, with ugly bounds on the integrals), which makes me hope for a solution method. However, I really cannot see how to go about it. Thus I ask, is there a method for computing the integral $(*)$?