jockstrap, it protects and get the job done.
Jockstraps are exposing, uncomfortable, chafe easily, and are otherwise obsolete. Compression shorts (w/ cup for protection) are super comfortable, they don't expose anything, can be worn as regular underwear, and many come with padding for legs. Unlike the nasty jockstrap, compression shorts keep the muscles warm and keep your muscles in place, lowering the chance of injury or sprains greatly.
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Compression shorts provide support for your thighs and glutes. But jockstraps do a better job supporting your genitals and keeping things from bouncing. They also tend to be cooler, since there's less material.
Depending how they fit, you might want to. Sometimes the legs are tight, but they don't really support your genitals - a jockstrap is designed to keep them in place better.