Error estimates with explicit constants for the tanh rule and the DE formula for indefinite integrals

Abstract

The tanh rule and the double-exponential (DE) formula are known as efficient quadrature rules for \emph{definite integrals} over a finite interval $(a, b)$. In this note we consider a numerical method for \emph{indefinite integrals} obtained by applying the tanh rule or the DE formula to the integration over the interval $(a, x)$ for each $x$. For these methods the conventional error analyses yield error estimates depending on $x$, which are impractical. We here present error estimates that do not depend on $x$, and furthermore, with explicit constants.