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%I #14 Jul 29 2021 11:05:52
%S 1,6,2,7,0,0,8,9,9,1,0,8,5,7,2,1,3,1,5,7,6,3,7,6,6,6,7,7,0,1,7,6,0,4,
%T 4,3,7,9,8,5,7,3,4,7,1,9,0,3,5,7,9,3,0,8,2,9,1,6,2,1,2,3,5,5,3,2,3,5,
%U 2,0,7,6,9,2,7,5,4,3,0,2,8,1,2,5,3,1,8,4,0,0,3,2,8,3,2,4,3,3,8,6,9,7,1,0,1
%N Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + sin(x)^2 + sin(y)^2) dy dx.
%F Equals -Pi^2*(log(2) + log(sqrt(2)-1)/2) + Pi * Integral_{x=0..Pi/2} log(1 + sqrt(1 + 1/(1 + sin(x)^2))) dx.
%F Equals limit_{n->infinity} Pi^2 * (log(A340396(n))/n^2 - log(2)) / 4.
%e 1.627008991085721315763766677017604437985734719035793082916212355323520769...
%t RealDigits[N[-Pi^2*(Log[2] + Log[Sqrt[2] - 1]/2) + Pi*Integrate[Log[1 + Sqrt[1 + 1/(1 + Sin[x]^2)]], {x, 0, Pi/2}], 120], 10, 110][[1]]
%Y Cf. A340396, A340322, A340350, A340422.
%K nonn,cons
%O 1,2
%A _Vaclav Kotesovec_, Jan 07 2021
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