The inverse tangent integral and its association with Euler sums (Q6562197)

The objective of this paper is to enhance the comprehension and usage of integrals in \[U_I^\pm(a, m, p, q, t)=\int_I \frac{x^a \log^p(x)\mathrm{Ti}_t(x^q)}{(1+\delta x)^{m+1}}dx, \] where \(a\in \mathbb{C}\backslash \mathbb{Z}_{\leq -2}\); \(m\in \mathbb{Z}_{\geq -1}\); \(t \in \mathbb{Z}_{\geq 0} \); \(p, q \in \mathbb{N}\); \(\delta=\pm 1\); \(I\) is the unit interval \((0, 1)\) or the positive real half line \((0, \infty)\), and \(\mathrm{Ti}_n(x)\) is the inverse tangent integral of order \(n\). For this purpose, the authors investigate those that involve integrands composed of logarithmic, inverse tangent, and other elementary function combinations. Then, they obtain a closed-form expression of the above integral using special functions, numbers, and constants, such as the Riemann zeta function and Euler harmonic numbers.