# How to evaluate $\int_0^1 {\log x \log(1-x) \log^2(1+x) \over x} \,dx$

Solve that the following integral: $$\int_0^1 {\log x \log(1-x) \log^2(1+x) \over x} \,dx.$$

I haven't solved it yet.

-
The is a definite integral, which has a numerical value, so in that sense, yes it is possible to express it in closed form, albeit depending on what you mean by closed form. –  pbs Nov 10 '13 at 9:28
$0.1099416891.$ –  user64494 Nov 10 '13 at 9:38
Notice that the integrand is undefined at $x=0$ and $x=1$ or at any $x$ between $0$ and $1$. –  user40615 Nov 10 '13 at 9:54
$$\ln(1-x)=\sum_{n=1}^\infty\frac{x^n}n$$ –  Lucian Nov 10 '13 at 10:50

You can find my solution here.

My argument here is greatly simplified by the use of complex analysis, thanks to the user Random Variable. So you may also want to check it, too.

-
Extremely impressive. Seriously, not enough adjectives to express how insanely great this solution is. –  Ron Gordon Nov 10 '13 at 13:30
@RonGordon Thank you for your praise! I was just lucky enough to find out that I can put things together... –  sos440 Nov 10 '13 at 13:44
Perhaps you should repost here in full. –  Ron Gordon Nov 11 '13 at 2:12
As a matter of fact, I've already seen a solution of true wonders. I've posted the question after several failures of trying to solve it in my own way. Thanks for your reply though. –  Leun Park Nov 11 '13 at 7:16
Wow. That is magnificent. Absolute masterpiece. Agree 100% with the people who suggest publishing. I've seen many, many papers in high ranking journals with far lesser quality. –  Bennett Gardiner Nov 11 '13 at 7:50