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Owing to the complexity of the exact calculation, synchrotron losses are usually estimated in system studies, with expressions derived from a plasma description using simplifying assumptions on the geometry, radiation absorption, and density and temperature profiles. In the present article, a complete formulation of the transport of synchrotron radiation is performed for realistic conditions of toroidal plasma geometry with elongated cross-section, using a quasi-exact method for the calculation of the absorption coefficients, and for arbitrary shapes of density and temperature profiles. The effects of toroidicity and temperature profile on synchrotron radiation losses are analysed in detail. In particular, when the electron temperature profile is almost flat in the plasma centre as, for example, in internal transport barrier confinement regimes, synchrotron losses are found to be much stronger than in the case where the profile is represented by its best generalized parabolic approximation, though both cases give approximately the same thermal energy content. Such an effect is not included in presently used approximate expressions. As an illustration, it is shown that in the case of an advanced high temperature plasma envisaged for a steady state commercial reactor, 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 of synchrotron radiation losses. This fit is derived from a large database which has been generated using a code implementing the complete formulation, and is optimized for massively parallel computing.