Copper tubs are handmade of malleable, natural metal. Because of this, they can sometimes develop high or low spots that prevent the bathwater from draining completely. There are a couple of things to try if you have a drainage issue with your copper tub.
First, ensure that the tub drain is not too tall, pushing the bottom of the tub upward from the floor. If this is the case, the drain will need to be lowered to create the proper slope for water to drain.
However, if the drain area is level to the floor, try covering the drain area with a towel and gently hammering it with a rubber mallet, pushing the drain area down. This will create a low spot near the drain and deter standing water.
You can use the same cloth and rubber mallet technique for other high spots in the tub that may be causing pooling. Additionally, you can do this on the underside of the tub if you have high areas that need to be brought back to level.