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We present IAS15, a 15th-order integrator to simulate gravitational dynamics.The integrator is based on a Gau\ss-Radau quadrature and can handleconservative as well as non-conservative forces. We develop a step-size controlthat can automatically choose an optimal timestep. The algorithm can handleclose encounters and high-eccentricity orbits. The systematic errors are keptwell below machine precision and long-term orbit integrations over $10^9$orbits show that IAS15 is optimal in the sense that it follows Brouwer's law,i.e. the energy error behaves like a random walk. Our tests show that IAS15 issuperior to a mixed-variable symplectic integrator (MVS) and other popularintegrators, including high-order ones, in both speed and accuracy. In fact,IAS15 preserves the symplecticity of Hamiltonian systems better than thecommonly-used nominally symplectic integrators to which we compared it.  We provide an open-source implementation of IAS15. The package comes withseveral easy-to-extend examples involving resonant planetary systems,Kozai-Lidov cycles, close encounters, radiation pressure, quadrupole moment,and generic damping functions that can, among other things, be used to simulateplanet-disc interactions. Other non-conservative forces can be added easily.

IAS15: A fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits