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Generalized log-sinh integrals
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Numbers
$n$
$k$
$x$ 
$\operatorname{Lsh}_n^{(k)}(x)$
2
0
$a$:
0.8316472421721863689616628700820596456998847230196935282056992132653494973365017092722424705626947784
2
0
$2a$:
0.9869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
2
0
$b$:
0.6885603792371556576720292986960330623610000339861255019932610311364092573515431512581030571477967464
3
0
$a$:
-1.920558570844575980359306539335704773936491706139814055426056102065234188949998177590788614444768818
3
0
$2a$:
-1.991089882371392968230758181371506714228628916274806756183866731846102530925464516611467853641321672
3
0
$b$:
-2.133836291964733045126740847998276306279356383350715642254543635531027805478607913990540727369073728
3
1
$2a$:
0.2404113806319188570799476323022899981529972584680997763584543110683676411572626180372911747218670516
3
1
$b$:
-0.2027971653867127485613947205175517647164536982627038277333890843450460671356801713472345319693650946
4
0
$a$:
5.954091854150286887022779289711324584956160165232850994974845445820501176222992800083460315506639357
4
0
$2a$:
5.990954427333662221085106748708180940427344354228792633526814798897901054422674888999883635498441967
4
0
$b$:
5.915236506143167754844778574177040243597185064234240308295572588017845210866340514389589325642086422
4
1
$2a$:
-0.2450752164550237958624617052561286374048717628812191953944903337648896550319136125399626878978259695
4
1
$b$:
-0.4661207128818308941799209621854503883862611169399606846900972446182191955940328953754716214035212911
4
2
$2a$:
0.1035880131109227799385749806249477468593384003349826370025881738402959057376265310749152711185280718
4
2
$b$:
-0.5656963978854031797569491107674444693840882395568473817233017422476954798083550983375992794764510230
5
0
$a$:
-23.96703691755914213829482919684496686791838099503370809999665649832504625142430195004920705675751147
5
0
$2a$:
-23.98782774996036739008086693933519050078158200014743696518691790050998809675781401313022772994955727
5
0
$b$:
-24.03037054035826138494263678343550627602321602349952895745467733775141688577711715883358932178540906
5
1
$2a$:
0.3712384672654671737861395334216768486191204049287745586798473225116780892888297525880361182837570599
5
1
$b$:
0.2510174008484959781125374477621693612970577267001533149035925103942885996256772604064229409523434008
5
2
$2a$:
-0.07097846975780870437600884496294466581226504002466154756722903803660887715960903406575991492979022369
5
2
$b$:
-0.4169883171974070579571371643981951543515167721411589146557937445353296386785284464541442504122109294
5
3
$2a$:
0.05635021092254895548199207679907508608731180763021678659351978144719442201928257319033614936053741190
5
3
$b$:
-0.9693157542805685947205473687493779060705150061190425803317676320613475453922113445809062700007662857
6
0
$a$:
119.9673019555723062187209955803653207843245894285143937960422482859615955727899098198845720557624957
6
0
$2a$:
119.9795991140767549617852804401899260295177634710678367913287476417972577767963041882919641936540323
6
0
$b$:
119.9547970146880617482244974167698416628577529306980274860720161993020688198945638720267943957886361
6
1
$2a$:
-0.7461917485527448824650290927246973747554456561042840751847088401460772239826674450673522670909394403
6
1
$b$:
-0.8148830477083279483516619888567767476038080527678457301288556971462895587945130719082892274975837033
6
2
$2a$:
0.07217812101045059169243780659423267293022775068444960651637610957743570625749114535983681149591197398
6
2
$b$:
-0.1201610646943565173591955726309151732585703538383183285341367068264648654905515344658204817602270588
6
3
$2a$:
-0.02913913084544511266108445805391049200990890624238116393117913847044386045958514351725272706875794984
6
3
$b$:
-0.5759880939613256056977312798893099243639111102366883214322925736233275858437998051295690792055422507
6
4
$2a$:
0.03482823759404336066830207968506789519067689562077743363797893061348368090558626742046699928487620242
6
4
$b$:
-1.557504324358268010108296880053671317439162755702770007542561349660849001932366815291169007208363335
7
0
$a$:
-719.9515467796277676055028434632198949376790915111168512655026830972305592317266606799615606481940446
7
0
$2a$:
-719.9590596202139214466857093815071116187750488057438630452200425028482207018652522897532957864256776
7
0
$b$:
-719.9738965971572269405645457924176495896131583132926120837652277033395806276534146396028407706548940
7
1
$2a$:
1.870199828822132168205061338333174424736721968166508287216706050533719912100562488978727721180034745
7
1
$b$:
1.829666179828968134965623511208500811842877846528274325115412850670517482193927752496528622538352049
7
2
$2a$:
-0.09722588143065943205624593577791867112734535939136179772991350576794162508647996239107132923808228304
7
2
$b$:
-0.2087671471310617170933973967290033844816614690269923262373243089061787568782176501809577219513466009
7
3
$2a$:
0.02235822126444696699747966318839605598315300776045682433901318839840920165202925036951698424154233948
7
3
$b$:
-0.2874664562894984464751896241415333896110887990336814132478143044427942749678600135783771157641024518
7
4
$2a$:
-0.01447734828657690779186411986146491775325896320432917916901049013774682597803221762830575637759009655
7
4
$b$:
-0.8861403107811902797093595874364662248361044341066912893360610139601258530060011129872963744465188864
7
5
$2a$:
0.02333715475542177649057299172093166327971884492288341802481715463712245377435911877043125758486874813
7
5
$b$:
-2.476990395500770208641809410550268417182117653732718882335541556381019168951894197561859070594000664
8
0
$a$:
5039.899577557626259571867449540290477350509421699560640338586375222812070713410200863542549123566705
8
0
$2a$:
5039.904276302947161035739404471695221280326990229116099080175543244435205547320042122903988629553551
8
0
$b$:
5039.895229950367944726145990084090333441467829243564127097490587522455040590742726859551341318021102
8
1
$2a$:
-5.617759170384834128028423876895963012426441562604387710525902195718360985803499128488555302056107510
8
1
$b$:
-5.642229564593522565019253800131257419461929563656597754623822965399364588723083318918732757453200852
8
2
$2a$:
0.1630527992138619472753825054788393179958270489812488554957873433688569862288547613035507047334135211
8
2
$b$:
0.09651871891098892119818331062927271013076059694153677568952997621993010506186487423431984421825359855
8
3
$2a$:
-0.02270564158077967813732695492040561694511622579926233801045472229069185735292069566682679009421907420
8
3
$b$:
-0.2047549448301742110938379297836101173316344109865096169102929773553026502052438649492076935256389957
8
4
$2a$:
0.008931773079893821298152833169927293149038230666121550183683420455624440396108632014943488622601728719
8
4
$b$:
-0.4931989956398286782469617008719150789498108565640894917278719022577779915026935844260119803994975078
8
5
$2a$:
-0.008114707521675741901087168421566990681372668452854259614487006038961468121240238226764444049974963894
8
5
$b$:
-1.408063411377850596410581151975763199752406521851016612316907305253748250051935092148630708078519397
8
6
$2a$:
0.01653419929773563634003750364149644293844804370566378737978495946584436685268126398248241761095460627
8
6
$b$:
-3.948724520344530848586005955021309512381071935830364684142568315110837646223940691903749103426526022
9
0
$a$:
-40319.74142927009794462399462670795722592683342048045934287169021242561216193037653830908644643542717
9
0
$2a$:
-40319.74442054083987547171045317673203109735587476388039503819980070674956106569770446941321889373914
9
0
$b$:
-40319.75001770181562462592431097586819794621530844174864652195712022166517257673327266619593428458549
9
1
$2a$:
19.67478057031755983704419907129563266816978992022348746227648001194041221042871132749320820043613800
9
1
$b$:
19.65975343323816255832363474239359385628775852479804080610371428906334773095875028574646853071690813
9
2
$2a$:
-0.3273359904593985059231485866275328514596091613236705995023983050458386175487355954528954555465180802
9
2
$b$:
-0.3678356730866058081353479941521499130769692105911493129322534196350895297352227328726337798648840004
9
3
$2a$:
0.02867906335248597199839190841239929403361046722113184888639852191958242491188976141971676846974462908
9
3
$b$:
-0.08096933876344377352470863488246599162946803981315909446306476692924791288713850039021304251321474167
9
4
$2a$:
-0.007290942679557105533988066539918383129982211726793464765618298413158534566060635971101913254398378020
9
4
$b$:
-0.3057939816052976834896657329690446862976356614928519768295565313283563841678813856239281663327271256
9
5
$2a$:
0.004189645173301493292546709192541526557502571889512208060002248534754677223023519856086656398881619898
9
5
$b$:
-0.8140788201735491001220774143547704159038155891655978802661962878884235330202842130978298988392160270
9
6
$2a$:
-0.004943023497333844909613798120976893150054837837919273397629746901799765115797589238359831132444119725
9
6
$b$:
-2.268475083657622685689653136209240353630972341421846200261906691121542170024525377424509143088782874
9
7
$2a$:
0.01220264016711193572363765627105741889789881726682141166425429454113860152939670636099025963261006308
9
7
$b$:
-6.331015849900948158887591236752442069195540640873699029179961985259185908623549372250887728088203071
10
0
$2a$:
362879.2326891642731792635531795716361269251172621649035376852384321551405717074016362858641138955453
10
0
$b$:
362879.2291855507773334651269777332080407184393710780033868171967356926483255815700083650754988250736
10
1
$2a$:
-78.72449651375718960835728557124820129040188806402911978651973871478626607999230185331597178214817909
10
1
$b$:
-78.73384779153496538986494875614082880061047186287516838195865867264542404008235616912176355334737430
10
2
$2a$:
0.7655257732792406630950193117598719638780259391162599158813262260268534471447939152079664367823570322
10
2
$b$:
0.7404926404783246305889834759299276837566282809312486086305693635692383651994599603831585915036419956
10
3
$2a$:
-0.04332141835230692445360359244506514084398393051998434445109185474200184604386933214718984160057709799
10
3
$b$:
-0.1105648812581632633991376049728589625127601581419534562939108273180443840940836832581889003649289969
10
4
$2a$:
0.007397974633128993376606652417320625020899306764939476860244072953547462057929583324170572804639885247
10
4
$b$:
-0.1739598692263947659724982818667037211293491661836708132703596091565171303075017913780378553475827186
10
5
$2a$:
-0.002861955582743845959176788588345743124054662671331956766711639601881466425421042365888850099626865915
10
5
$b$:
-0.4943631203769579481064629070949917172428467629605506656391245814640183175529894216249539259919498777
10
6
$2a$:
0.002195305267610830290775221208143281760391112489757555854572580472156317398791047490431677338267562705
10
6
$b$:
-1.337820529963928801641643090874972581616757190245350859691883169670396652491360071818913004041741334
10
7
$2a$:
-0.003200125415510869677089491192890770185270724731118583036617402891406780678527324389278777789735552373
10
7
$b$:
-3.684881949712787808984229795534490607810305337160209690690580957496984905989339243544814800303164649
10
8
$2a$:
0.009291309973000719216487978490923542524623680242579404515142939305351417556722184942365502821790034725
10
8
$b$:
-10.21506689821101749989869480354147269684484351003051507531439507895989909398408412325011265125263748
Definition
The generalized log-sinh integral is $\operatorname{Lsh}_n^{(k)}(x)=-\int_0^x t^k\log^{n-1-k}(2\sinh(t/2))\,dt$, with natural logarithms [1], equation (33). For $x>0$, integers $n\geq1$ and $0\leq k<n$, this is an ordinary convergent improper integral.
Parameters
$n$
—   total order ($n\geq1$)
$k$
—   integration-variable moment order ($0\leq k<n$)
$x$
—   positive real endpoint ($x>0$)
Formulas
(1)
$\operatorname{Lsh}_n^{(n-1)}(x)=-x^n/n$.
(2)
$\operatorname{Lsh}_2^{(0)}(x)=\pi^2/6-\operatorname{Li}_2(e^{-x})-x^2/4$. Here $\operatorname{Li}_2(z)=\sum_{j\geq1}z^j/j^2$ for $|z|<1$. Differentiation and the limit at $x=0$ give this identity.
(3)
$\operatorname{Lsh}_2^{(0)}(2a)=\pi^2/10$ and $\operatorname{Lsh}_3^{(1)}(2a)=\zeta(3)/5$, using the dilogarithm identity and evaluation in [1], Example 11. Here $\zeta$ is the Riemann zeta function.
(4)
For integers $s\geq3$, $\operatorname{Lsh}_s^{(1)}(2a)=\dfrac{(s-2)!}{(-2)^{s-2}}F_s(-1)$, where $F_s(u)=\sum_{j\geq1}u^j/(j^s\binom{2j}{j})$ is the inverse binomial sum. This follows from repeated integration of $uF_s'(u)=F_{s-1}(u)$, starting with $F_2(-4\sinh^2(x/2))=-x^2/2$, and using $A(2a)=0$.
(5)
For integers $r\geq0$, $\operatorname{Lsh}_{r+1}^{(0)}(2x)=2\sum_{j=0}^r\binom rj H_{j+1,r-j+1}(x)$, where $H_{m,n}(x)=-\int_0^x A(t)^{m-1}C(t)^{n-1}\,dt$ and $C(t)=\log(2\cosh(t/2))$ [2], Definition 2.1. This follows by substitution from $A(2t)=A(t)+C(t)$; in particular, $H_{m,1}(x)=\operatorname{Lsh}_m^{(0)}(x)$.
Comments
(6)
Put $\varphi=(1+\sqrt5)/2$, the golden ratio, $a=\log\varphi$, and $b=2\log(1+\sqrt2)=2\operatorname{arsinh}(1)$. The endpoint keys $\mathtt{logphi}$, $\mathtt{2logphi}$ and $\mathtt{2logsilver}$ denote exactly $a$, $2a$ and $b$. All radicals and logarithms use their positive real branches.
(7)
The exponent $k$ is the moment order and $n-1-k$ is the logarithmic power. Write $\operatorname{Lsh}_n=\operatorname{Lsh}_n^{(0)}$ for the basic integrals. The leading minus sign, the factor $2$ and the half-argument are retained; no factorial is absorbed into the stored value. The logarithm itself can be negative, and its absolute value is not taken.
(8)
The kernel $A(t)=\log(2\sinh(t/2))$ has its unique positive zero at $2a$, the alternating Apéry endpoint of [1], Example 11. The smaller endpoint $a$ occurs in the harmonic-weighted binomial reductions of [2], Lemma 2.2 and section 4; its basic-moment slice supplies the boundary terms of the duplication formula. The endpoint $b$ is a bounded exploratory reference: $u=-4\sinh^2(x/2)$ gives $u=-4$ there, the negative convergence endpoint of the inverse-binomial family [3]. It also includes integration across the zero of $A$. No separate application is asserted for every higher moment at $b$.
(9)
Near $t=0$, the logarithm is $\log t+O(t^2)$, so its finite powers multiplied by $t^k$ are integrable. The elementary edge $k=n-1$ is given by a formula and is omitted from the numerical selection.
Programs
(P1)
Python (mpmath)
import mpmath as mp

def log_sinh(n, k, endpoint, dps=140):
    with mp.workdps(dps):
        phi = (1 + mp.sqrt(5))/2
        x = {"logphi": mp.log(phi),
             "2logphi": 2*mp.log(phi),
             "2logsilver": 2*mp.asinh(1)}[endpoint]
        if n < 1 or not 0 <= k < n:
            raise ValueError("require n>=1 and 0<=k<n")
        def f(s):
            t = x*mp.exp(-s)
            return -x**(k+1)*mp.exp(-(k+1)*s)*mp.log(2*mp.sinh(t/2))**(n-1-k)
        return mp.quad(f, [0, 1, 4, 16, 64, 256, mp.inf])

def selected_arguments():
    for n in range(2, 11):
        for k in range(n-1):
            for x in ("logphi", "2logphi", "2logsilver"):
                if x != "logphi" or (k == 0 and n <= 9):
                    yield n, k, x

log_sinh(3, 1, "2logphi")
# Quadrature is an independent numerical check; generate.py supplies bounds.
References
[1]
J. M. Borwein and A. Straub, Special Values of Generalized Log-sine Integrals (2011), section 5.2, equation (33), Theorem 4 and Example 11. (arXiv)
[2]
S. Charlton, H. Gangl, L. Lai, C. Xu and J. Zhao, On two conjectures of Sun concerning Apery-like series (2022), Definition 2.1, Lemma 2.2 and section 4. (arXiv)
[3]
A. I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums (2004), equations (1.1)-(1.3) and section 3. (arXiv)
Links
Similar tables
Generalized log-sine integrals use the trigonometric kernel and real-angle endpoints; they are the trigonometric counterpart discussed in [1].
Inverse binomial sums supply the alternating first-moment relation (4); the higher moments here are distinct quantities.
Riemann zeta values include the value entering (3).
Polylogarithms at rational arguments give related special-function context. The algebraic arguments in (2) are not its current rational-argument family.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every $2\leq n\leq10$, $0\leq k\leq n-2$ at $x=2a,b$, and $k=0$, $2\leq n\leq9$ at $x=a$, for 98 entries. The triangle covers all non-elementary moments through a noncanonical order cutoff of ten at the intrinsic golden endpoint and the exploratory inverse-binomial boundary. The shorter basic slice at $a$ supplies the mixed-family duplication boundaries through logarithmic degree eight. These conditional selections are documented in argument-selection.md)
How they were obtained:

All 98 values carry 100 significant digits from real ball arithmetic. Expanding $B(t)=\log(2\sinh(t/2)/t)$ by the infinite product [4] gives coefficients $(-1)^{j+1}\zeta(2j)/(j(2\pi)^{2j})$. The generator integrates a finite polynomial in $t^2$ and $\log t$ exactly in form, using ball coefficients.

more

With $q=(x/(2\pi))^2<1/10$, the omitted kernel tail after $N$ terms is bounded by $2q^{N+1}/((N+1)(1-q))$; a difference-of-powers bound is integrated and added to every result. Endpoint radicals and logarithms are enclosed and their errors propagated. At 100 requested digits, $N=150$ and 614-bit working arithmetic retain at least 547 bits of relative accuracy. The derivation is attached as computation.md. Independent 140-digit mpmath quadrature checks every value, with 175-digit repeats at four difficult cases; the checker also tests the dilogarithm formula and every stored alternating first-moment identity. Known-answer and deliberately wrong controls validate the checker. The attached executable checks and results distinguish these numerical checks from the rigorous error bound.



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