Fast and reliable symplectic integration for planetary system $N$-body problems

We apply one of the exactly symplectic integrators, that we call HB15, of \cite{HB15}, along with the Kepler problem solver of \cite{WH15}, to solve planetary system $N$-body problems. We compare the method to Wisdom-Holman methods (WH) in the \texttt{MERCURY} software package, the \texttt{MERCURY} switching integrator, and others and find HB15 to be the most efficient method or tied for the most efficient method in many cases. Unlike WH, HB15 solved $N$-body problems exhibiting close encounters with small, acceptable error, although frequent encounters slowed the code. Switching maps like \texttt{MERCURY} change between two methods and are not exactly symplectic. We carry out careful tests on their properties and suggest they must be used with caution. We then use different integrators to solve a 3-body problem consisting of a binary planet orbiting a star. For all tested tolerances and time steps, \texttt{MERCURY} unbinds the binary after 0 to 25 years. However, in the solutions of HB15, a time-symmetric Hermite code, and a symplectic Yoshida method, the binary remains bound for $>1000$ years. The methods' solutions are qualitatively different, despite small errors in the first integrals in most cases. Several checks suggest the qualitative binary behavior of HB15's solution is correct. The Bulirsch-Stoer and Radau methods in the \texttt{MERCURY} package also unbind the binary before a time of 50 years, suggesting this dynamical error is due to a \texttt{MERCURY} bug.