Lets say you invest $1billion in a city in recession. The investment will have a multiplier, boosting the value of your investment. Thus if keynesianism was correct there would be an increased incentive to invest during recessions
Yes, the big success of Keynesianism was uncovering that there is an incentive to invest in a demand-led recession, from the perspective of the whole economy, because of the multiplier.
Just as you say.
However, there is no incentive to invest for individual profit-maximising firms, for exactly the reason dismalscience gives: the investment stimulates general demand; but (in general) very little demand of that newly-stimulated demand is for the particular good or service being invested in.
It is this gap between the sum of preferences of all maximisers of individual utility, and the preferences of the sum of all of those actors, that Keynesian investment fills.
Your question includes a flawed assumption: why would you assume that you'd recapture any significant portion of the additional output? Let's say that the multiplier is .05, so that for your billion dollar investment (let's say the you build a new factory), others in the city increase their total spending by \$50 million. What fraction of that \$50 million do you suppose would be spent on the products produced by your billion-dollar factory?
If you're being even halfway realistic, you should assume it's 1% or less— so maybe you'd net an additional half-million dollars on an investment you made in a factory that you built for a billion dollars during a recession... when people are generally purchasing less than usual.
Suffice it to say, Keynesianism does not imply that this would be a good investment decision.