Owing to the complexity of the exact calculation,synchrotron losses are usually estimated in system studies, with expressionsderived from a plasma description using simplifying assumptions on thegeometry, radiation absorption, and density and temperature profiles. In thepresent article, a complete formulation of the transport of synchrotronradiation is performed for realistic conditions of toroidal plasma geometrywith elongated cross-section, using a quasi-exact method for the calculationof the absorption coefficients, and for arbitrary shapes of density andtemperature profiles. The effects of toroidicity and temperature profile onsynchrotron radiation losses are analysed in detail. In particular, when theelectron temperature profile is almost flat in the plasma centre as, forexample, in internal transport barrier confinement regimes, synchrotron losses are found to be muchstronger than in the case where the profile is represented by its bestgeneralized parabolic approximation, though both cases give approximately thesame thermal energy content. Such an effect is not included in presently usedapproximate expressions. As an illustration, it is shown that in the case of anadvanced high temperature plasma envisaged for a steady state commercialreactor, synchrotron losses represent approximately 20% of the total losses,so that this term becomes significant in the power balance of such a plasma.Finally, the authors propose a seven variable fit for the fast calculation ofsynchrotron radiation losses. This fit is derived from a large database whichhas been generated using a code implementing the complete formulation, and isoptimized for massively parallel computing.