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Abstract

In this paper, we evaluate the following families of definite integrals in closed form and we show that they are expressible only in terms of the dilogarithm function and the inverse tangent integral, and elementary functions. \begin{equation*} \int_{0}^{1}\frac{\log\big(x^m+1\big)}{x+1}\thinspace{\rm d}x \quad \mbox{and}\quad \int_{0}^{1}\frac{\log\big(x^m+1\big)}{x^2+1}\thinspace{\rm d}x, \end{equation*} where $m$ is a positive odd integer. When $m$ is a positive even integer, these integrals have been evaluated previously by Sofo and Bat{\i}r, and the case where $m$ is an odd integer has been left as open problems. The integrals of the first kind arise in Zagier's work on the Kronecker limit formula. In addition, we demonstrate that a functional equation satisfied by the Herglotz-Zagier-Novikov function is a very specific case of of a more general formula, and give numerous illustrative examples.