<p>Reaction rate determination is an essential problem in the theories of cosmological and stellar nucleosynthesis. I investigate the closed-form analytic evaluation of thermonuclear reaction rates, which are necessary for both cosmic and celestial nucleosynthesis. An alternate velocity distribution to the Maxwell-Boltzmann distribution can be considered in nuclear fusion processes occurring in the interior of stars when there is a diversion from hydrostatic equilibrium. The extension of non-resonant reaction rate integrals in standard, cut-off, depletion, and screening instances is examined in this study, utilising the MacDonald distribution. This study is primarily on the use of generalized special functions that represent the extended non-resonant thermonuclear functions in mathematical physics. Moreover, an attempt is made to analyse the reaction probability integrals corresponding to the several forms of the slowly varying cross-section factor S(E).</p>