A sequence of elementary integrals related to integrals studied by Glaisher that contain trigonometric and hyperbolic functions

TL;DR

This paper generalizes integrals originally studied by Glaisher, incorporating trigonometric and hyperbolic functions, and applies these generalizations to derive an analog of a known integral by Ismail and Valent.

Contribution

It introduces a new class of integrals extending Glaisher's work and connects them to existing integrals through novel generalizations.

Findings

New generalized integrals involving trigonometric and hyperbolic functions

An analog of Ismail and Valent's integral derived

Enhanced understanding of integral relationships in special functions

Abstract

We generalize several integrals studied by Glaisher. These ideas are then applied to obtain an analog of an integral due to Ismail and Valent.