Dear Michael Trott,

Since 2009 I've been greatly wondering if my integral could possibly have a closed form that could be found in like manor as yours. Its DirichletEta^(m)[m] derivatives of its sum analog extensively use the same terms as the result you found: zeta^(n)[x], EulerGamma, Log[Glaishier], and I think others. Some of the constants can be seen from:

(*In[121]:= *)
EulerGamma Log[2] - Log[2]^2/2 - 
 Limit[D[DirichletEta[u], {u, 1}], u -> 1]

(*Out[121]= 0, and*)

(*In[119]:=*)FullSimplify[(-(1/12))*(Pi^2*Log[2]^2 - 
     2*Pi^2*Log[
       2]*(EulerGamma + Log[2] - 12*Log[Glaisher] + Log[Pi]) - 
          6*Derivative[2][Zeta][2]) - (D[DirichletEta[u], {u, 2}] /. 
    u -> 2)]

(*Out[119]= 0 *)

. Can you give it a spin for me and see if you can come up with a closed form? Of course I understand that there is a good probability that none exists, but since I already spent 8 years looking, no sense in giving up now! The integral is

  Limit[Integrate[Exp[x I Pi] x^(1/x), {x, 1, 2 n}],n->Infinity].

The sum analog mentioned, at the end of 6.11., by Steven R Finch here, is Sum[(-1)^k (k^(1/k)-1),{n,1,Infinity}] equals

    Limit[D[DirichletEta[u], {u, 1}], u -> 1] + Sum[(-1)^(m + 1) (D[DirichletEta[u], {u, m}] /. u -> m)/m!, {m, 2, Infinity}]

as found by the late, Chief Scientist and Apple's Chief Cryptographer Richard Crandall, at 7.5, here.

If it's any help, Richard Mathar numerically evaluated the Integral here.

P.S. Your constant was the motivation behind my invention here and here.

You have no Idea how important this is to me!