If you trust your packing and box making capabilities, then when looking for cheapest, you have to go to the shipping calculators for every shipper and see what it cost to ship that size package and weight.

When I was shipping a bike several years ago Fed Ex came out to be the cheapest. I played with the dimensions of the package and found significant savings by shortening the length of my box. Which was easy to do since I made my own box.

When packing your bike though, don't think that putting words like "fragile" or "this end up" mean anything to the people handling it. I pack expecting someone to throw it across the room.