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I'll go out on a limb and predict that if you do the integration carefully,
ALL of the capacitor's original energy will go into Ohmic heating. The
differential equation whose integral you are using I(t) does not know of
any radiation mechanism; it states Kirchoff's loop law which says energy
is totally accounted for by the terms in the equation. You would have to
include a term which sinks energy by way of some "radiation resistance".
Math is not magic; neither is it a source of new physical information; it
is logic, and its conclusions will only make explicit what is already
implicitly in the premises which it was handed.