Notes and Remarks on certain logarithmic integrals

TL;DR

This paper revisits logarithmic integrals involving rational functions, providing a method to evaluate integrals of the form ∫₀¹ ln(ln(1/x)) R(x) dx, enhancing understanding of these complex integrals.

Contribution

It introduces a new approach to evaluate a class of logarithmic integrals with rational functions, expanding existing methods for such integrals.

Findings

Derived explicit evaluation techniques for integrals of the form ∫₀¹ ln(ln(1/x)) R(x) dx

Provided formulas to compute these integrals efficiently

Enhanced understanding of the properties of logarithmic integrals involving rational functions

Abstract

Logarithmic integrals revisited. We consider integrals of the form $\int_0^1 \ln{\ln{(\frac{1}{x})}}R{(x)}{\rm d}x$ again, where $R{(x)}$ is a rational function, and we will explain a way to obtain their values.