Compact Coefficient Formulae for Logarithmic Tangent and Hyperbolic Integrals

Abstract

We develop compact coefficient-extraction formulae for several families of hyperbolic, logarithmic tangent, and Malmsten-type integrals whose values are finite linear combinations of odd zeta values and even Dirichlet beta values. The principal advantage of these formulae is that coefficients previously encoded by recursive arrays or nested finite sums are replaced by a single coefficient of an explicit elementary expression. This makes the coefficients easier to compute, keeps the dependence on the parameters visible, and reveals structural features---such as vanishing ranges, extremal coefficients, and sign patterns---without hidden cancellations. For shifted hyperbolic integrals with numerator $\sinh((2k+1)x)$, the coefficients are expressed through Chebyshev--arcsine extractions. The same mechanism yields Laurent coefficient formulae for logarithmic tangent integrals and leads to direct proofs of simple initial and terminal coefficients, including a parity-free terminal identity. For $m,n\geq1$, $m\geq n$, and $m+n$ even, we prove \[ \int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n} (2^{2p+1}-1) \frac{\zeta(2p+1)}{\pi^{2p}} [u^{m+n-2p}](u\cot u)^{m+1}. \] In the diagonal case, this gives the family $\int_0^\infty(\tanh x/x)^N\,dx$ in a direct, non-recursive form and makes the disappearance of the initial zeta values immediate.