Fast computation of close-coupling exchange integrals using polynomials in a tree representation. The semi-classical atomic-orbital close-coupling method is a well-known approach for the calculation of cross sections in ion-atom collisions. It strongly relies on the fast and stable computation of exchange integrals. We present an upgrade to earlier implementations of the Fourier-transform method.For this purpose, we implement an extensive library for symbolic storage of polynomials, relying on sophisticated tree structures to allow fast manipulation and numerically stable evaluation. Using this library, we considerably speed up creation and computation of exchange integrals. This enables us to compute cross sections for more complex collision systems.