The imaginary parts will cancel.

ii = Integrate[(1/(1 + Exp[-x]))*Log[1/(1 + Exp[-x])], x]

(* Out[359]=Log[-E^x] Log[1/(1 + E^x)] + 1/2 Log[1/(1 + E^x)]^2 - 
 Log[1/(1 + E^x)] Log[E^x/(1 + E^x)] - PolyLog[2, 1 + E^x] *)

hi = Limit[ii, x -> Infinity]
lo = Limit[ii, x -> -Infinity]

(* Out[360]= -(\[Pi]^2/3)

Out[361]= -(\[Pi]^2/6) *)

hi - lo

(* Out[362]= -(\[Pi]^2/6) *)

The output contains functions with complex values, but the end result is real:

int = Integrate[(1/(1 + Exp[-x]))*Log[1/(1 + Exp[-x])], x]
int2 = FullSimplify[ComplexExpand[int], Element[x, Reals]]
FullSimplify[ComplexExpand[int] == int2, Element[x, Reals]]
Plot[int - int2, {x, -1, 1}]

Unfortunately we cannot check the derivative of int2, because int2 contains Re, which is not differentiable in the complex sense.