Symbolic Integration in Weierstrass-like Extensions

This paper studies the integration problem in differential fields that may involve quantities reminiscent of the Weierstrass $\mathrm{\wp }$ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of $\mathrm{\wp }$.