I am interested in finding $$f(n) = \int_0^1\int_0^1\ldots\int_0^1\frac{1}{1-\prod_{i=1}^{n}\ln(x_i)}\,dx_1dx_2\ldots dx_n$$ for positive integer $n$.
For example, $$f(2)=\int_0^1\int_0^1\frac{1}{1-\ln(x_1)\ln(x_2)}\,dx_1dx_2$$
I found $f(1) = e \cdot E_1(1) \approx 0.596$, $f(2)$ diverges, $f(3) \approx 0.724$. $$$$ My questions
How can I find $f(n)$ for positive integer $n$?
If that is not possible, what is an approximation?
I know for sure that if $n$ is odd, $f(n)$ converges. This is because $\frac{1}{1-\prod_{i=1}^{n}\ln(x_i)}$ will converge for $0 \le x_i \le 1$. I also suspect that if $n$ is even, $f(n)$ diverges.