True value of an integral in Gradshteyn and Ryzhik's table

TL;DR

This paper corrects a known integral from Gradshteyn and Ryzhik's tables by evaluating it explicitly in terms of elliptic integrals using advanced analysis techniques.

Contribution

It provides the true value of a previously incorrect integral entry, expressed through elliptic integrals, with a detailed derivation involving real and complex analysis.

Findings

Corrected the value of the integral from Gradshteyn and Ryzhik.

Expressed the integral in terms of elliptic integrals of the first and third kind.

Demonstrated a standard but involved evaluation method using advanced analysis.

Abstract

Victor Moll pointed out that entry 3.248.5 in the sixth edition of Gradshteyn and Ryzhik tables of integrals was incorrect. He asked some years ago what was the true value of this integral. I evaluate it in terms of two elliptic integrals. The evaluation is standard but involved, using real and complex analysis. I prove $$\int_0^{\infty}\frac{dx}{(1+x^2)^{3/2}[\varphi(x)+\sqrt{\varphi(x)}]^{1/2}}= \frac{\sqrt{3}-1}{\sqrt{2}}\Pi(\pi/2,k,3^{-1/2})-\frac{1}{\sqrt{2}}F(\alpha, 3^{-1/2}),$$ where $\varphi(x)=1+4x^2/2(1+x^2)^2$, $k=2-\sqrt{3}$, $\alpha=\arcsin\sqrt{k}$ and $F(\varphi,k)$, $\Pi(\varphi,n^2,k)$ the elliptic integrals of the first and third kind respectively.