Integrals containing the infinite product $\prod_{n=0}^\infty\Bigl[1+\bigl(\frac{x}{b+n}\bigr)^3\Bigr]$

TL;DR

This paper investigates integrals involving an infinite product with a cubic term, deriving closed-form evaluations and new formulas related to these integrals.

Contribution

It introduces new integral formulas involving infinite products with cubic terms, expanding the analytical understanding of such integrals.

Findings

Closed-form evaluation of a specific integral as 1/3

Derivation of additional integral formulas involving infinite products

Extension of methods to integrals with similar structures

Abstract

We study several integrals that contain the infinite product ${\displaystyle\prod_{n=0}^\infty}\left[1+\left(\frac{x}{b+n}\right)^3\right]$ in the denominator of their integrand. These considerations lead to closed form evaluation $\displaystyle\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$ and to some other formulas.