  We analytically as well as numerically disclose the effects of the higher
order correction terms in the gravity and in the gauge field on the properties
of $s$-wave holographic superconductors. On the gravity side, we consider the
higher curvature Gauss-Bonnet corrections and on the gauge field side, we add a
quadratic correction term to the Maxwell Lagrangian. We show that for this
system, one can still obtain an analytical relation between the critical
temperature and the charge density. We also calculate the critical exponent and
the condensation value both analytically and numerically. We use a variational
method, based on the Sturm-Liouville eigenvalue problem for our analytical
study, as well as a numerical shooting method in order to compare with our
analytical results. For a fixed value of the Gauss-Bonnet parameter, we observe
that the critical temperature decreases with increasing the nonlinearity of the
gauge field. This implies that the nonlinear correction term to the Maxwell
electrodynamics make the condensation harder. We also study the holographic
conductivity of the system and disclose the effects of Gauss-Bonnet and
nonlinear parameters $\alpha$ and $b$ on the superconducting gap. We observe
that for various values of $\alpha $ and $b$, the real part of conductivity is
proportional to the frequency per temperature, $\omega /T$, as frequency is
enough large. Besides, the conductivity has a minimum in the imaginary part
which is shifted toward greater frequency with decreasing the temperature.
