What are the best options to repair this kind of damage in the bathtub

Most bathtubs that aren't cast iron or steel are acrylic plastic.

If the crack is right through the tub and water is leaking though, you should replace the tub. If it's an emergency and you need a short-term repair, I'd use a two-part epoxy. Choose a thin epoxy (2:1 ratio) from the selection at the home center and use a putty knife to try to force it into the crack. Let it cure for 24 hours. This will be at best a short term solution because if tub cracked it must be because it flexed in that spot - it will flex again and the brittle epoxy repair will crack again quickly.

If the crack is a surface scratch, it can be repaired with a repair kit such as offered by http://www.surface-repair.com/. The problem is color matching: your tub will have changed color over time and it will be difficult or impossible to get a good match.

Another approach is using fine sandpaper. You can get very fine wet-and-dry sandpaper at an auto parts store like Napa. Get 320, 600, 1200 and 2000 grit. Apply some elbow grease to the scratch with the 320. Try to keep the scratches localized and in one direction. Use the 320 until the scratch is gone. Then move to the 600 and keep it up until the scratches from the 320 are gone. By the time you're done 1200 you may be satisfied, but if not you should be able to get a fairly glossy finish with the 2000.

Cast iron tubs are very heavy and expensive. Rust is not a problem with cast iron tubs as long as the enamel top layer is intact (if you get a chip in it, you can get enamel repair kits). My wife did not like the old cast iron tub in our bathroom because it was cold when standing in it during a shower and made bath water get cold fast. Evidently she was not taking baths in the “cast iron method” which is to put super-hot water into the bathtub and allow the heat to transfer into the tub, this will cool the water down but keep it at a reasonable temperature longer (in theory at least, YMMV).

Enameled steel it basically a cheaper version of the cast iron, it is a lighter because the metal is much thinner. I doubt the “hot bath trick” described above for cast iron tubs will work in a steel tub. Like cast iron tubs, you should not have to worry about a steel tub rusting or cracking.

Acrylic (plastic) tubs come in a different quality levels. Some are really cheap, thin plastic that have the potential to crack over time (particularly if they are poorly supported/improperly installed). However some acrylic tubs use thicker stronger plastic (often with fiber glass embedded for tensile strength) that are very sturdy.

I recently had to make this decision; we replaced our cast iron tub with plastic Kohler model. I knew I did not want to lug a cast iron tub up to the 2nd story. I thought about a steel tub but they are still 120-150 pounds. I was worried about getting a cheap plastic one that would break so I settled on a $700 plastic Kohler tub that is pretty thick.