This paper introduces efficient and accurate algorithms for simulating the rotation of a three-dimensional rigid object and compares them to several prior methods. First, we introduce a second-order-accurate method that incorporates a third-order correction; then we introduce a third-order-accurate method; and finally we give a fourth-order-accurate method. These methods are single-step and the update operation is only a single rotation. The algorithms are derived in a general Lie group setting. Second, we introduce a near-optimal energy-correction method which allows exact conservation of energy. This algorithm is faster and easier to implement than implicit methods for exact energy conservation. Our third-order method with energy conservation is experimentally seen to act better than a fourth-order-accurate method. These new methods are superior to naive Runge-Kutta or predictor-corrector methods, which are only second-order accurate for sphere-valued functions. The second-order symplectic McLachlan-Reich methods are observed to be excellent at approximate energy conservation but are not as good at long-term accuracy as our best methods. Finally we present comparisons with fourth-order-accurate symplectic methods, which have good accuracy but higher computational cost.