A hybrid approach for the integration of a rational function (Q1196865)

A hybrid algorithm is proposed which gives an indefinite integral for a rational function \(q/r\) with floating-point but real coefficients. It consists of four steps: (1) approximate square-free decomposition of a polynomial \(r\); (2) numerical root-finding of the equation \(r=0\); (3) partial fraction decomposition of \(q/r\); (4) transformation of partial fractions into the indefinite integral. The approximate-GCD algorithm and the Durand-Kerner root-finding method are used in the first and second step. Coefficients of partial fractions are determined by the residue theory. Here symbolic differentiation is used. Partial fractions are then transformated into the indefinite integral by using simple rules of integrals. Some arguments are listed for the effectiveness of the hybrid method. Comparisons of definite integrals for some examples obtained by hybrid, symbolic and some numerical methods are given.