f = ((1 - x - y + x y + x Log[x] - x y Log[x] + y Log[y] - x y Log[y] + x y Log[x] Log[y]) Log[1 + x y])/((1 - x) x (1 - y) y Log[x] Log[y])

fi = Integrate[f, {x, 0, 1}, {y, 0, 1}]

NIntegrate[f, {x, 0, 1}, {y, 0, 1}]

(* Out[11]= 0.158253 *)

{x -> Exp[-u], y -> Exp[-v]}

{u, 0, [Infinity]}, {v, 0, [Infinity]}

ff = (1/u - 1/(E^u - 1)) (1/v - 1/(E^v - 1)) Log[1 + E^(-u - v)]

NIntegrate[ff, {u, 0, [Infinity]}, {v, 0, [Infinity]}]

0.158253

(-1)^(k+1)/k (-(1/(-1 + E^u)) + 1/u) (-(1/(-1 + E^v)) + 1/v) E^(-k u -k v)

fi = Sum[(-1)^(k + 1)/k g[k]^2, {k, 1, [Infinity]}]

g[k_] := Integrate[(-(1/(-1 + E^u)) + 1/u) E^(-k u), {u, 0, [Infinity]}]

g[k] = HarmonicNumber[k] - EulerGamma - Log[k]

Table[g[k], {k, 1, 5}]

(* {1 - EulerGamma, 3/2 - EulerGamma - Log[2],
 11/6 - EulerGamma - Log[3], 25/12 - EulerGamma - Log[4],
 137/60 - EulerGamma - Log[5]} *)

NSum[(-1)^(k + 1)/k g[k]^2, {k, 1, [Infinity]}]

(* 0.158253 *)