After some integration process, I ended up with the following expression:
(1/(b (-1 + E^b) Re[b]))E^-Re[b]( b E^b - b + E^Re[b] Re[b] - E^(b + Re[b]) Re[b] + E^Re[b] Sqrt[E^(-2 b) (-1 + E^b)^2] Re[b] + b E^(b + Re[b]) Sqrt[E^(-2b)(-1 + E^b)^2] Re[b] )
all is good, but this expression is supposed to be equal to
1 + (2/b) e^(-b) - 1/b
via simple numerical trials, i can confirm that they are equal. But, it would be great if I can actually make Mathematica simplify that nasty expression into this innocent form.
Fullsimplify, it does not work. Does anyone have any suggestion?