Many problems in biological fluid-structure interaction have been studied with Peskin's immersed boundary method (IBM). This method defines a relatively simple mathematical modeling framework, and allows for the use of standard fluid solvers. However, when evaluated computationally in many applications, a very small time step is required. This time step is not restricted by accuracy, but stability, causing the temporal problem to be stiff. Previous attempts to address the stability restriction of IBM use semi or fully implicit schemes that require the code, including the fluid solver, to be rewritten, and these have yet to be implemented in application focused studies. Here, new ideas for addressing the computational cost of IBM are presented, which rely on a novel method for splitting the spatial field into stiff and non-stiff components. With this splitting, the impact on the largest stable time step and computational cost for stiff problems is investigated through multi-implicit, multirate and time parallel techniques. All of the algorithms presented here improve upon the stability restriction of IBM, whether with a larger stable time step or overall speedup. In addition, they are focused on the treatment of the immersed boundary and no changes to the fluid solver are necessary, making them more accessible to applications.