An elementary evaluation of a quartic integral

Abstract

We provide an elementary evaluation for the integral

\[N_{0.4}(a; m) = \int_{0}^{\infty} \frac{dx}{(x^4 + 2ax^2 + 1)^{m+1}}\]

where \(m \in \mathbb{N}  and  a \in (-1, \infty)\)

\[N_{0.4}(a; m) = \frac{2^{m+3/2}}{(a+1)^{m+1/2}}\pi P_m(a)\]

\({for } P_m(a) \text{ a polynomial in } a.\)