calculus - How to compute the following integral in $n$ variables?

How can the following integral be calculated:

$$I_n=\int_0^1\int_0^1\cdots\int_0^1\frac{\prod_{k=1}^{n}\left(\frac{1-x_k}{1+x_k}\right)}{1-\prod_{k=1}^{n}x_k}dx_1\cdots dx_{n-1}dx_n$$
There should be $n$ integral signs, but I didn't know how to write that.

It is easy to show that $I_1=\ln(2)$. After partial fractioning and the help of Wolfram Alpha, I managed to show that $I_2=4\ln(2)-2\ln^2(2)-\frac{\pi^2}{6}$.

But how to derive a general result? Any help would be highly appreciated!

Edit:

As a supplementary question, how to calculate this slightly modified integral:

$$J_n=\int_0^1\int_0^1\cdots\int_0^1\frac{\prod_{k=1}^{n}\left(\frac{1-x_k}{1+x_k}\right)}{1+\prod_{k=1}^{n}x_k}dx_1\cdots dx_{n-1}dx_n$$
Again, it can be shown easily, that $J_1=1-\ln(2)$.