The new data of the elastic scattering of $^{6}$He+$^{12}$C at about 40 MeV/nucleon are analyzed in the eikonal approximation. The $^{6}$He+$^{12}$C phase-shift function is evaluated completely without any {it ad hoc} assumption by a Monte Carlo integration, which makes it possible to use a realistic 6-nucleon wave function for a halo nucleus $^{6}$He. The effect of the breakup of $^6$He on the elastic differential cross sections as well as the optical potential is studied at different energies from 40 to 800 MeV/nucleon. PACS number(s): 24.10.-i; 21.60.Ka; 25.60.Bx; 25.10.+s Keywords: Eikonal; Glauber; Monte Carlo; Halo; Breakup
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