1  Divide and Conquer

The Oscillatory method to Mathematica's NIntegrate solves this, once the integral is transformed using Lambert's W function (also known as ProductLog) to ∫_0^∞ (cos u)/(u (1 + 1/(W (u))))   u. This method evaluates the integral between zeroes and extrapolates the partial sums to the limit. One can get more by summing an initial segment by hand and then using the method just described on the tail. Our technique can be used to obtain 60 digits. Bornemann, using fancier techniques to accelerate convergence, obtained 100 digits. Boersma and Jansen used contour integration to transform the integral to two integrals that can be easily computed numerically. The first 100 digits:
0.3233674316777787613993700879521704466510466257254696616810364434317903372106728944319303704641024514

NIntegrate[Cos[u]/(u (1 + 1/ProductLog[u])), {u, 0, ∞}, MethodOscillatory]

0.323367

Boersma & Jansen transformed the integral to: ∫_0^(π/2) sin (t   sin t) _ (_ (_)) ... + ∫_1^∞ cos (π/2  y)    y^(-y - 1)   y. The second integral takes some effort and I am able to get 300 digits out of this approach.

Created by Mathematica  (June 27, 2004)