  We introduce a class of fourth order symplectic algorithms that are ideal for
doing long time integration of gravitational few-body problems. These
algorithms have only positive time steps, but require computing the force
gradient in additional to the force. We demonstrate the efficiency of these
Forward Symplectic Integrators by solving the circularly restricted three-body
problem in the space-fixed frame where the force on the third body is
explicitly time-dependent. These algorithms can achieve accuracy of
Runge-Kutta, backward time step and corrector symplectic algorithms at step
sizes five to ten times as large.
