Triggers after insert

I seem to be observing that on a trigger I've created to fire
AFTER INSERT, if I intentionally introduce some sort of syntax error to
the trigger for example, and insert a row, the row does not appear in
the table, even though I've declared it to fire after insert.  Shouldn't
that row be there regardless of whether the trigger does or does not
fire successfully?  Or is this currently not how it works?  The docs
seemed to indicate that it would work that way.

The end result is that I need to make sure the insert happens whether or
not the trigger is successful.

Thanks,
Fran