I came across the following question which asked
Prove that $$\int_{0}^{1} \prod_{k=1}^{\infty} (1-x^k)=\frac{4\pi\sqrt{3}\sinh{\frac{\pi\sqrt{23}}{3}}}{{\sqrt{23}\cosh{\frac{\pi\sqrt{23}}{2}}}}$$
Note: This is the one of the first integrals I have seen like this. I have been looking up other questions (similar to mine) and am still unable to figure out a method to try.
Attempt: I first examined different cases for $k$ and I am able to compute those integrals fine, since they are just polynomials when expanded.
However, I am still stuck with where to begin.
$$\int_{0}^{1} \prod_{k=1}^{\infty} (1-x^k)=(1-x)(1-x^2)(1-x^3) \cdots (1-x^n) \cdots$$
If I were to factor a $-1$ from each term I can see that each term, after the first, contains a factor of $(x-1)$ which might help with some kind of substitution, but that is about as far as I got.
If someone could please offer a hint that will help me get going with this problem.
Thank you.
