Semi-Analytic Trajectory Analysis of Light in Generic Static Spacetimes

We study a unified semi-analytical framework to study null geodesics and weak-field light deflection in generic static, spherically symmetric spacetimes of the form $d{s}^2=-\alpha (r,\delta )d{t}^2+\gamma (r,\delta )d{r}^2+\beta (r,\delta )d{\mathrm{\Omega }}_2^2,$ where $\alpha$, $\beta$, and $\gamma$ encode model-dependent deviations from Schwarzschild gravity inspired from [Phys.Rev.D 112 (2025) 12, 124072]. Starting from the exact first-order orbit equation, we derive a compact master equation for the impact-parameter-dependent trajectory $u(\phi )\equiv 1\mathrm{/}r(\phi )$ and obtain a model-independent expression for the bending angle $\alpha (b)$ in terms of generic metric functions and their derivatives. This master equation is then solved semi-analytically by three complementary techniques: (i) the homotopy perturbation method (HPM), (ii) the variational iteration method (VIM), and (iii) a calibrated impulse (single-kick) approximation expressed directly in terms of the effective gravitational potential. As nontrivial test beds we consider a scalar-hairy Reissner-Nordström-like black hole where the scalar hair enters as $Q_s$ in an effective charge parameter. Then we derive closed-form expressions for the deflection angle, identify the leading scalar-hair, and compare the accuracy and convergence properties of HPM, VIM, and the impulse method against the standard Schwarzschild limits. Our results show that the generic formulation in $(\alpha ,\beta ,\gamma )$ can efficiently accommodate a broad class of modified gravity black hole solutions. We further supplement the analytic treatment with a compact numerical results against the exact null-geodesic integral near the photon sphere in order to delineate the practical range of validity of the three approximation schemes.