We study the problem of efficient integration of variational equations in multidimensional Hamiltonian systems. For this purpose, we consider a Runge-Kutta-type integrator, a Taylor series expansion method and the so-called "Tangent Map" (TM) technique based on symplectic integration schemes, and apply them to the Fermi-Pasta-Ulam beta (FPU-beta) lattice of N nonlinearly coupled oscillators, with N ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique - which shows the best performance among the tested algorithms - and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.