magnetobremsstrahlung radiation The magnetobremsstrahlung force $f^i$ acting on a particle is proportional to 
$$f^i\propto\left[\frac{d^2 u^i}{ds^2}-u^iu^k\left(\frac{d^2u_k}{ds^2}\right)\right],$$
where the second term could be rewritten in the form $(F_{kl}u^l)(F^{km}u_m)u^i$ that is a standard form for the radiated $4$-momentum or radiated energy for $i=0$ (that comes from the expression $d{\cal E}=(2e^2/3c^3)w^2dt$, where $w$ is a particle acceleration and ${\cal E}$ is a particle energy. 
Energy loss could be found also by integrating  $f^i\,ds$ over the world line. After integration the first term of $f^i$ goes to zero because $du^i/ds=0$ at infinity and the second term gives obviously the standard radiation form $(F_{kl}u^l)(F^{km}u_m)u^i$. 
I am wondering why do we have to integrate over the whole particle world line (or may be average the energy losses over time) to find the correct expression for the radiated energy? Where in determining energy losses by the general expression $d{\cal E}=(2e^2/3c^3)w^2dt$ we neglected the first term  $(d^2u^i/ds^2)$ of the radiation force $f^i$? Is it coming from the point that $d{\cal E}=(2e^2/3c^3)w^2dt$ is derived for the wave zone, i.e. far from the charge? If yes, how to express this explicitly?
What about the energy loss rate at a given point of the world line (it should take into account the full expression for $f^i$)? 
 A: The Larmor formula gives EM energy per unit time that flies through big sphere centered in accelerated charged body and whose surface is far away from the particle, where the EM field is almost equal to the wave component of the retarded field of the charged body.
Based on this formula, people tried to infer a consistent expression for radiation reaction force acting on the charged body and the LAD formula is one such attempt. However, all these attempts have problems with consistency so the resulting LAD expression for radiation reaction force can only be approximate.
If you are interested in the energy radiated by a macroscopic charged body (antenna), use the Larmor formula and ignore the LAD formula, it is not needed.
If you are interested in the radiation reaction force acting back on the body or the motion of the radiating charges, you can use the LAD force as approximation to the actual EM self-force. For exact value of this force, there is (as far as I know) no useful theory, apart from "add all the internal forces that act in the charged body". Lorentz and Abraham did some approximate calculations on those lines for charged spheres and arrived at infinite series of terms; the first one gives modification of the effective mass, the second one is the LAD force, proportional to rate of change of acceleration. The other terms are usually neglected, but they are there and they reflect further details of the internal motions in the charged body and their effect on the actual self-force.
