Compact Coefficient Formulae for Logarithmic Tangent and Hyperbolic Integrals

Abstract

We give compact coefficient formulae for several hyperbolic integrals whose values are linear combinations of odd zeta values and even Dirichlet beta values. First, the coefficients occurring in shifted integrals with numerator \(\sinh((2k+1)x)\) are rewritten as single Chebyshev-arcsine coefficient extractions. For \(k=0\), these identities recover the zeta- and beta-type integrals studied by Kyrion, but replace recursive coefficients by the coefficients of powers of \(\arcsin x\). The same method also yields compact coefficient formulae for logarithmic tangent integrals. Finally, for \(m,n\geq1\), \(m\geq n\), and \(m+n\) even, we prove the general formula \[ \int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n} (2^{2p+1}-1) \frac{ζ(2p+1)}{π^{2p}} [u^{m+n-2p}](u\cot u)^{m+1}. \] In the diagonal case this gives the family \(\int_0^\infty(\tanh x/x)^N\,dx\) in a non-recursive cotangent coefficient form and makes the initial vanishing of the zeta expansion immediate.