Six minke whales (BaLaenoptera acutorostrata) were instrumented with VHF‐radio transmitters and four with sonic speed‐depth transmitters off the west coast of northern Norway and Svalbard and followed within view for up to 24 h. During such periods their respiratory rate was continuously recorded and their energy expenditure estimated according to Folkow&Blix (1992) at different swimming speeds and types of activity. We found that cost of swimming is remarkably low in these large animals and that their estimated daily energy expenditure on average only amounts to 80 kJ kg‐1