Adaptive integration of functions maintaining a constant sign of the sixth derivative

This paper introduces an adaptive numerical integration method specifically designed for functions of class $$C^6[a,b]$$ with a sixth derivative of constant sign. The approach is based on a refined inequality involving the three-point Gauss quadrature ( $$\mathcal {G}$$ ) and the four-point Lobatto quadrature ( $$\mathcal {L}$$ ). We utilize a specific linear combination, $$Q = \frac{3}{4}\mathcal {G} + \frac{1}{4}\mathcal {L}$$ , and demonstrate that for functions with a sixth derivative of constant sign, the approximation error is effectively controlled by the difference between these two quadratures. We provide a rigorous justification for the stopping criterion of the adaptive algorithm. Numerical experiments, including the approximation of the integrals of a reciprocal function as well as the exponential function, show that the proposed method significantly outperforms existing adaptive techniques designed for lower-order convexity, requiring substantially fewer subintervals to achieve high precision (up to $$10^{-16}$$ ).