  We have analytically investigated the effects of non-linearity on the free
energy and thermodynamic geometry of holographic superconductors in $2+1
-$dimensions. The non-linear effect is introduced by considering the coupling
of the massive charged scalar field with Born-Infeld electrodynamics. We then
calculate the relation between critical temperature and charge density from two
different methods, namely, the matching method and the divergence of the scalar
curvature which is obtained by investigating the thermodynamic geometry of the
model. The two results are slightly different from numerical values but the
effects of non-linearity gets captured in our analysis.
