On the generalized inverse tangent integral and Catalan's constant

TL;DR

This paper introduces new identities for the inverse tangent integral by linking it to dilogarithmic functions and auxiliary arctangent integrals, providing explicit formulas and representations.

Contribution

It develops a systematic approach connecting arctangent integrals, Catalan's constant, and polylogarithmic functions through generating-function analysis.

Findings

Derived explicit formulas for inverse tangent integrals.

Connected arctangent integrals to dilogarithmic structures.

Produced new integral representations involving polylogarithms.

Abstract

In this paper, we develop new identities for the inverse tangent integral by connecting it to the dilogarithmic (polylogarithmic) structure and to a carefully designed auxiliary arctangent integral $Ti_2(a)$ with a tunable endpoint. The core idea is based on the introduction of an auxiliary integral depending on two parameters and analyzing it via a generating-function perspective. This converts the integral into an explicit formula, yielding a compact representation in terms of the real part of a dilogarithmic expression plus a companion dilogarithm contribution. In parallel, the inverse tangent integral is rewritten through standard integral transformations into a form governed by the imaginary part of a dilogarithm evaluated at a complex argument, producing a clean polylogarithmic description. Overall, we establish a coherent bridge between arctangent-type integrals, identities connected to Catalan's constant, and the systematic generation of new integral representations and decompositions.