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To distinguish between regular and chaotic orbits in Hamiltonian systems, the Global SymplecticIntegrator (GSI) has been introduced [Libert et al., 2010], based on the symplectic integrationof both Hamiltonian equations of motion and variational equations. In the present contribution,we show how to compute efficiently the MEGNO indicator jointly with the GSI. Moreover, wediscuss the choice of symplectic integrator, in fact we point out that a particular attention has tobe paid to the structure of the Hamiltonian system associated to the variational equations. Theperformances of our method is illustrated through the study of the Arnold diffusion problem.Keywords: Symplectic integration, chaos, variational equations, MEGNO, Arnold diffusion.