An Ahmed-like integral

Abstract

The so-called Ahmed integral $$ \int_{0}^{1}\frac{\arctan\left(\sqrt{2+x^{2}}\right)}{(1+x^{2})\sqrt{2+x^{2}}}\,\mathrm{d} x=\frac{5\pi^{2}}{96}, $$ has attracted considerable interest since its appearance in the "American Mathematical Monthly" in 2001. Several proofs and extensions have been proposed, including a probabilistic multivariate approach introduced by Pla based on powers of the Gaussian integral. In this note, we extend Pla's method to the fifth power of the Gaussian integral. By expressing this power as a sequence of iterated integrals and performing successive reductions, we obtain a new integral identity closely related to Ahmed's integral. In particular, we prove that $$ \int_{0}^{1}\frac{\arctan\!\left(\sqrt{\displaystyle\frac{2+x^2}{4+x^2}}\right)}{(1+x^{2})\sqrt{2+x^{2}}}\,\mathrm{d} x=\frac{\pi^2}{30}. $$ The derivation suggests that Pla's technique can systematically generate a family of Ahmed-type integrals associated with higher powers of the Gaussian integral.