Numerous technical innovations are based on micromagnetic processes. A popular example are read/write operations on classical hard disk drives. There, bits are encoded as magnetization patterns in granular ferromagnetic components. The micromagnetic phenomena describing these dynamics are usually modeled by the Landau-Lifshitz-Gilbert (LLG) equation, if we assume a homogeneous temperature distribution in the medium. Analytical solutions to this highly nonlinear partial differential equation with the non-convex constraint of a constant magnetization modulus are in general unknown. Therefore, the development and the analysis of numerical methods for the approximate solution of the LLG equation are of great interest. In this work, we consider time-marching schemes based on the tangent plane approach. These integrators rely on first-order finite element spaces for the spatial discretization and a predictor-corrector time integration to preserve the modulus constraint. The first explicit version of a tangent plane integrator was proposed in 2006 and generalized to an unconditionally convergent integrator, which treats the highest-order term implicitly in time, in 2008. Recently, in [Alouges et al. (Numer. Math., 128, 2014)] the first formally almost second-order in time accurate tangent plane integrator was presented. There, for the improved order of convergence lower-order terms have to be integrated implicitly in time - in particular, this includes the stray field approximation of the magnetostatic Maxwell equations, whose evaluation requires high computational effort. Due to the non-locality of the stray field, the corresponding finite element system-matrix is dense. Therefore, in practice rather than assembling this matrix, the linear system is solved iteratively. In this diploma thesis, we show that after the first time step, by using an Adams-Bashforth-like two-step method, linear self-adjoint terms can be integrated explicitly in time. In particular, our analysis covers the stray field computation by hybrid FEM-BEM methods. Ultimately, by using this implicit-explicit (IMEX) approach we derive the first tangent plane integrator, which is almost second-order accurate in time, converges unconditionally to a weak solution of LLG, and is efficient in the sense that, besides one stray field computation, one only requires the direct solution of one sparse linear system per time step. Numerical experiments, simulated with the LLG solver implemented in the course of this diploma thesis, confirm the results of our analysis.