Finding $\int_0^1\int_0^1\ldots\int_0^1\frac{1}{1-\prod_{i=1}^{n}\ln(x_i)}\,dx_1dx_2\ldots dx_n$

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.

• Can you give some background about this integral? Like how did you encounter it and why is it in your interested? Because it doesn't look easy. – Zacky Aug 7 '19 at 8:34
• @ㄴㄱ It just popped into my head after seeing another question that was similar. Unfortunately, I don't remember what that question was. – automaticallyGenerated Aug 7 '19 at 17:15