At a birth vs At birth vs At the birth

at a birth—during a single act of reproduction:

• Cats sometimes have four or five young at a birth.

at birth—at the time when a baby or an animal is born:

• No animal at birth is mature or perfect in intelligence.

at the birth—

1. present or attending during the birth:

• Josephine did not want her husband at the birth of her own children.

2. during the emergence of smth.:

• The UN had been the indispensable midwife at the birth of their country.