A256127 Decimal expansion of the second Malmsten integral: Integer_{x >= 1} log(log(x))/(1 + x + x^2) dx, negated.

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

Offset

0,2

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

Formula

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

Equals Integral_{x>=0} log(x)/(1 + 2*cosh(x)) dx.

Equals Pi*(8*log(2*Pi) - 3*log(3) - 12*log(Gamma(1/3)))/(6*sqrt(3)).

Example

-0.12632148170620903636522675325320239184424430946528...

Maple

evalf(Pi*(8*log(2*Pi) - 3*log(3) - 12*log(GAMMA(1/3)))/(6*sqrt(3)), 120); # Vaclav Kotesovec, Mar 17 2015

Mathematica

RealDigits[Integrate[Log[Log[1/x]]/(1 + x + x^2), {x, 0, 1}], 10, 100][[1]] (* Alonso del Arte, Mar 16 2015 *)
RealDigits[Pi*(8*Log[2*Pi] - 3*Log[3] - 12*Log[Gamma[1/3]])/(6*Sqrt[3]), 10, 105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

Prog

(PARI) Pi*(8*log(2*Pi) - 3*log(3) - 12*log(gamma(1/3)))/(6*sqrt(3)) \\ Michel Marcus, Mar 18 2015
(PARI) intnum(x=0, 1, log(log(1/x))/(1 + x + x^2))
(PARI) intnum(x=1, oo, log(log(x))/(1 + x + x^2))
(PARI) intnum(x=0, [oo, 1], log(x)/(1 + 2*cosh(x))) \\ Gheorghe Coserea, Sep 26 2018

Crossrefs

Cf. A115252 (first Malmsten integral), A256128 (third Malmsten integral) , A256129 (fourth Malmsten integral), A073005 (Gamma(1/3)), A256165 (log(Gamma(1/3))), A061444 (log(2*Pi)), A002391 (log 3), A002194 (sqrt 3).

Sequence in context: A108443 A201761 A011042 * A136758 A056113 A251755

Adjacent sequences: A256124 A256125 A256126 * A256128 A256129 A256130

Keyword

nonn,cons

Author

Iaroslav V. Blagouchine, Mar 15 2015

Status

approved