on simplifying an expression

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. I tried, Fullsimplify, it does not work. Does anyone have any suggestion?

Answer

Algebraic simplifications like Simplify and FullSimplify can be used with the second argument - assumptions. We can assume e.g. that b is a real number i.e. b ∈ Reals (otherwise the system assumes that b is complex) :

Simplify[ (1/(b (-1 + E^b) Re[b])) E^-Re[b](-b + b E^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^(-2 b) (-1 + E^b)^2] Re[b]),  b ∈ Reals] // 
 TraditionalForm

enter image description here

Since there are two cases b >= 0 and b < 0 (in general there might be more cases depending on the assumptions) we should map Expand on the output ( common shorthands Map -- /@ and MapAll -- //@)

Expand //@ % // TraditionalForm

enter image description here

The same answer you can get with FullSimplify.

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