A Note on Certain Elliptical and Pseudo-Elliptical Integrals

In the determination of the general solution of a partial differential equation, unless all the integrals arising in Lagrange’s characteristic method can be effected analytically, the entire exercise becomes meaningless. In this research note, we describe one particular integration involving elliptical integrals for which an exact evaluation is required. All the four known special cases of the integral are shown to arise from the same integral formula involving the inverse cosh function. Some general theory describing a logarithmic-polynomial approach is presented, and which is only partially successful in the present context. For the particular evaluation under investigation the logarithmic-polynomial approach is only effective in a special case, which is nevertheless a new result and may prove an important comparison for any future analytical or numerical investigations. However, all the known exact analytical evaluations may be couched in the logarithmic approach which is based upon the inverse cosh function.