  The Gibbons-Werner (GW) method is a powerful approach in studying the
gravitational deflection of particles moving in curved spacetimes. The
application of the Gauss-Bonnet theorem (GBT) to integral regions constructed
in a two-dimensional manifold enables the deflection angle to be expressed and
calculated from the perspective of geometry. However, different techniques are
required for different scenarios in the practical implementation which leads to
different GW methods. For the GW method for stationary axially symmetric (SAS)
spacetimes, we identify two problems: (a) the integral region is generally
infinite, which is ill-defined for some asymptotically nonflat spacetimes whose
metric possesses singular behavior, and (b) the intricate double and single
integrals bring about complicated calculation, especially for highly accurate
results and complex spacetimes. To address these issues, a generalized GW
method is proposed in which the infinite region is replaced by a flexible
region to avoid the singularity, and a simplified formula involving only a
single integral of a simple integrand is derived by discovering a significant
relationship between the integrals in conventional methods. Our method provides
a comprehensive framework for describing the GW method for various scenarios.
Additionally, the generalized GW method and simplified calculation formula are
applied to three different kinds of spacetimes--Kerr spacetime, Kerr-like black
hole in bumblebee gravity, and rotating solution in conformal Weyl gravity. The
first two cases have been previously computed by other researchers, affirming
the effectiveness and superiority of our approach. Remarkably, the third case
is newly examined, yielding an innovative result for the first time.
