The study:

A force turns around the center O in a circle (dotted lines), the force keeps its angle with the radius (beta is constant = 30°). The radius of the circle is 1. The force is 1. An external device gives the force F. It's easy to calculate the work needed to rotate the force from 0° (vertical) to 90° it is 0.5 from vertical to horizontal (rotate of 90°).

Now, take the black arm. The black arm turns around the red point, but it takes the force F at the end and it must increase its length if I want the force F turns around the center O.

Now the work is :

There is a difference !

Maybe I need to do that:

But there is always a difference:

2*cos(x)*sin(x+pi/6)-2*sin(x)*cos(x+pi/6) dx from pi/4 to 0

I found my error, it was the length in the integration. The sum of energy is 0.