A256128 Decimal expansion of the third Malmsten integral: int_{x=1..infinity} log(log(x))/(1 - x + x^2) dx, negated.

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

Offset: 0

Author

Iaroslav V. Blagouchine, Mar 15 2015

Keywords

• nonn
• cons

Examples

			-0.671719601885874542354405069288779884008802066219356...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Iaroslav V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. PDF file
• Wikipedia, Carl Malmsten

Crossrefs

Cf. A115252 (first Malmsten integral), A256127 (second Malmsten integral), A256129 (fourth Malmsten integral), A073005 (Gamma(1/3)), A002162 (log 2), A002391 (log 3), A053510 (log Pi), A002194 (sqrt 3).

Programs

Maple

evalf(Pi*(7*log(2)+8*log(Pi)-3*log(3)-12*log(GAMMA(1/3)))/(3*sqrt(3)),120); # Vaclav Kotesovec, Mar 17 2015

Mathematica

RealDigits[Pi*(7*Log[2]+8*Log[Pi]-3*Log[3]-12*Log[Gamma[1/3]])/(3*Sqrt[3]),10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

PARI

Pi*(7*log(2)+8*log(Pi)-3*log(3)-12*log(gamma(1/3)))/(3*sqrt(3)) \\ Michel Marcus, Mar 18 2015

Formula

Equals integral_{x=0..1} log(log(1/x))/(1 - x + x^2) dx.

Equals integral_{x=0..infinity} log(x)/(1 - 2*cosh(x)) dx.

Equals Pi*(7*log(2) + 8*log(Pi) - 3*log(3) - 12*log(Gamma(1/3)))/(3*sqrt(3)).