Does invariance under infinite small transformation imply invariance to the finite one?

Let’s say that I have finite chiral transform and I would like to show invariance of Dirac’s Lagrangian when $m=0$ under it.

The chiral transform is defined as:
$$psi(x) rightarrow psi'(x) =e^{i alpha gamma_5} psi(x)$$
where the Dirac’s Lagrangian:
$$L = bar psi (x) (i hbar gamma^{mu} partial_{mu} – 0c) psi(x)$$

If I consider infinitsmall transformation of above:
$$psi(x) rightarrow psi'(x) =(1 + i alpha gamma_5) psi(x)$$
I obtain that Lagrangian transforms as:
$$L rightarrow L’ = L + O(alpha^2)$$
where $O(alpha^2)$ is term with with order $alpha^2$. Is it enough to say that Lagrangian is invariant under finite transformation?