A Unified Substitution Method for Integration

Abstract

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 $e^{\pm i\arccos(y)}$ and $e^{\pm i\operatorname{arcsec}(y)}$ 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. For several natural families, cancellation between the parametrized radical and the Jacobian makes the transformed integrand a finite Laurent polynomial, so it can be integrated term by term. We recover Euler's first and second substitutions from these transforms after the necessary scaling, sign choices, and component-dependent reciprocal reparametrizations and provide worked examples; in particular, the classical Weierstrass substitution is obtained as a local direct corollary of Transform 5. The framework also yields Lobachevsky's formula $\tan(\Pi(u)/2)=e^{-u}$ as the positive hyperbolic specialization of one core identity, with an equivalent derivation through the reciprocal coordinate $y=\operatorname{sech}u$. Binomial sum-and-difference identities streamline the back-substitution of paired Laurent powers such as $p^n\pm p^{-n}$. Under the reported computational protocol, all tested instances for Transforms 2--5 passed the reported derivative/numerical checks, as did 94 of 100 instances for Transform 1. Selected benchmark families also exhibited reduced expression swell or improved completion coverage relative to the native Wolfram Language function $\texttt{Integrate}$ under the reported protocol, although these results concern the targeted test families and do not establish general superiority.