There is a proven way to eliminate body odour, but it is one that (rather surprisingly) most people seem to shy away from or at least overlook/ignore. My interpretation is that this is because it has nothing to do with the cosmetics and other industries (in fact is likely to have them in howls of protest) and is more or less totally free while being spectacularly effective. It is also (or perhaps therefore) not at all "cool".
It comes in two separate parts. The first is to accept that the fungi and bacteria which produce the smells not only live on our skin but also in our clothes - unless we kill nearly all of them off. Remember that fungi and bacteria are everywhere and they are great survivors - most of them are harmless if not beneficial. Thus, every time we put on clothes that have previously been worn, we re-awaken fungi/bacteria in/on the fabric by providing moisture and warmth. It is entirely normal to give off both of these and we should not interfere with our normal functions (as anti-perspirants do). Clothes, bedding, towels, etc. should periodically be washed at 60 degrees or higher (if "sour" then, as a campaing, several times in a row after only short use) to rid them of most of the populations. It is important to dry them quickly, preferably outdoors where daylight and ozone have a very beneficial effect. Letting them "simmer" on a clothes rack indoors gives fungi/bacteria another chance.