A388042 Decimal expansion of Integral_{x>=e} log(log(x))/(1+x^2) dx.

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

Offset

0,1

Comments

Integral_{x=0..1/e} log(log(x))/(1+x^2) is complex number 0.2152489028... + i*1.107453576148368... = A388042 + i*A393630.

Integral_{x=1/e..1} log(log(x))/(1+x^2) is complex number -0.475691709... + i*1.35994752412397... = -A393691 + i*A390125.

Integral_{x=0..1} log(log(x))/(1+x^2) is complex number -0.2604428063... + i*2.467401100272... = A115252 + i*A091476.

Integral_{x>=0} log(log(x))/(1+x^2) is complex number -0.5208851637565... + i*2.467401100272... = -2*A115252 + i*A091476.

Integral_{x=1..e} log(log(x))/(1+x^2) is real negative number -0.47569170911830734... = -A393691.

Integral_{x>=e} log(log(x))/(1+x^2) is real positive number 0.215248902817318895... = A388042.

Links

Sean A. Irvine, Table of n, a(n) for n = 0..999

Formula

Equals real part of Integral_{x=0..1/e} log(log(x))/(1+x^2).

Equals Integral_{x=arctan(e)..Pi/2} log(log(tan(x))) dx.

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

Let A = 0.215248902817318895... B = -0.47569170911830734... then

A+B = (Pi/4) log(4*Pi^3/Gamma(1/4)^4) = -0.260442806300988...

Equals Sum_{k>=0} (-1)^(k+1) * Ei(-(2*k+1)) / (2*k+1). - Sean A. Irvine, Mar 04 2026

Example

0.21524890281731889512316472...

Maple

evalf(Int(log(log(x))/(1+x^2), x = exp(1)..infinity), 105); # Vaclav Kotesovec, Feb 24 2026

Mathematica

RealDigits[NIntegrate[Log[Log[x]]/(1 + x^2), {x, E, Infinity}, WorkingPrecision -> 110], 10, 105][[1]]
(* or *)
NSum[(-1)^(k + 1) ExpIntegralEi[-(2*k + 1)]/(2*k + 1), {k, 0, Infinity}, WorkingPrecision -> 105]

Crossrefs

Cf. A091476, A115252, A390125, A393630, A393691.

Sequence in context: A093664 A298212 A087620 * A253809 A262213 A379219

Adjacent sequences: A388039 A388040 A388041 * A388043 A388044 A388045

Keyword

nonn,cons

Author

Artur Jasinski, Feb 22 2026

Status

approved