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Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2, z=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2) dz dy dx.
8

%I #24 Feb 28 2023 23:47:41

%S 6,4,8,5,6,9,6,4,6,5,2,1,8,4,9,7,6,9,3,7,0,8,5,8,1,3,7,2,1,0,3,3,1,5,

%T 7,6,4,1,5,2,2,6,6,3,2,5,6,1,7,9,7,6,3,1,6,8,3,1,7,3,8,8,4,2,4,5,2,5,

%U 5,5,2,3,8,7,8,4,0,0,5,6,7,8,5,4,1,8,8,8,8,7,6,4,1,9,6,8,5,7,5,5,3,9,1,7,4

%N Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2, z=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2) dz dy dx.

%C Integral_{x=0..Pi/2, y=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2) dy dx = G*Pi, where G is Catalan's constant A006752.

%H Vaclav Kotesovec, <a href="/A340322/b340322.txt">Table of n, a(n) for n = 1..200</a>

%F Equals limit_{n->infinity} Pi^3 * log(A340182(n)) / (8*n^3).

%F Equals Pi^3 * log(2)/8 + Integral_{x=0..Pi/2, y=0..Pi/2, z=0..Pi/2} log(3 + cos(2*x) + cos(2*y) + cos(2*z)) dz dy dx.

%e 6.485696465218497693708581372103315764152266325617976316831738842452555238784...

%p evalf(Integrate(log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2), x = 0..Pi/2, y = 0..Pi/2, z = 0..Pi/2));

%o (PARI) intnum(x = 0, Pi/2, intnum(y = 0, Pi/2, intnum(z = 0, Pi/2, log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2)))) \\ 20 valid digits

%Y Cf. A071763, A097469, A340182, A340350, A340421, A340422.

%K nonn,cons

%O 1,1

%A _Vaclav Kotesovec_, Jan 04 2021