  We present compact analytic expressions for 3-flavor neutrino oscillation
probabilities with invisible neutrino decay, where matter effects have been
explicitly included. We take into account the possibility that the oscillation
and decay components of the effective Hamiltonian do not commute. This is
achieved by employing the techniques of inverse Baker-Campbell-Hausdorff (BCH)
expansion and the Cayley-Hamilton theorem applied in the 3-flavor framework. If
only the vacuum mass eigenstate $\nu_3$ decays, we show that the treatment of
neutrino propagation may be reduced to an effective 2-flavor analysis in the
One Mass Scale Dominance (OMSD) approximation. The oscillation probabilities
for $P_{\mu\mu}$, $P_{ee}$, $P_{e\mu}$ and $P_{\mu e}$ -- relevant for reactor,
long baseline and atmospheric neutrino experiments -- are obtained as
perturbative expansions for the case of only $\nu_3$ decay, as well as for the
more general scenario where all components of the decay matrix are non-zero.
The analytic results thus obtained match the exact numerical results for
constant density matter to a high precision and provide physical insights into
possible effects of the decay of neutrinos as they propagate through Earth
matter. We find that the effects of neutrino decay are most likely to be
observable in $P_{\mu\mu}$. We also point out that at any long baseline, the
oscillation dips in $P_{\mu\mu}$ can show higher survival probabilities in the
case with decay than without decay, and explain this feature using our analytic
approximations.
