A Unified Substitution Method for Integration

We present a branch-consistent framework for integrals involving quadratic radicals by expressing exponentials of principal inverse trigonometric functions in algebraic form. Two identities for $e^{\pm i{\cos }^{-1}(y)}$ and $e^{\pm i{\sec }^{-1}(y)}$ on principal branches yield five explicit substitution templates that map common radicals and half-angle composites to rational functions of a single parameter. The resulting differentials are independent of the sign choice once the branches are fixed, reducing domain bookkeeping across circular and hyperbolic regimes. We recover Euler’s first and second substitutions from these transforms up to trivial reparametrizations and provide worked examples; in particular, the classical Weierstrass substitution is obtained as a direct corollary of Transform 5. A binomial-difference identity streamlines back-substitution terms such as $t^n-t^{-n}$. A CAS benchmark of 100 integrals indicates improved predictability and reduced expression swell relative to general-purpose integration routines.