Indefinite Integral for -.5*(ln(2))^2

[SumThePrimes]

Hello, I have recently encountered an integral that I have been able to evaluate in a sick, unholy way, and for making a proof much more elegant I would like a simple way to evaluate the integral from 0 to infinity of ln(x)/(e^x+1) . thank you!

 

[Millennial]

I haven't tried them but maybe differentiation under the integral sign (set 1 to any arbitrary number) and contour integration?

 

[flyintiger]

Is the denominator you have listed e^(x+1) or (e^x)+1? Just want to verify before trying the problem.

 

[SumThePrimes]

Is the denominator you have listed e^(x+1) or (e^x)+1? Just want to verify before trying the problem.

It is (e^x)+1 . I tried using limits on the bounds and then integration by parts and then switching certain values to other limited integrals and canceling... I tell you what that didn't work. I also tried a limit for the natural logarithm with an integral for the Dirichlet eta function, but that gave me a very tough looking limit that I had no chance with.

edit: The limit was lim x->+1 (\Gamma(x)\eta(x)-ln(2))/(x-1) , if anyone has some cool trick for that.