Is learning to speak Danish worth it?

Learning to speak Danish can be difficult, even if you speak its close linguistic cousins, English and German. While the written language isn’t too tough to figure out, the spoken language is a headache. Danes pronounce only small bits of each word and smash those small bits together. Even the Swedes and Norwegians have trouble understanding spoken Danish. If you’re only in Denmark for a short time, is it worth it to learn more than just the basic pleasantries in Danish?