A Unified Substitution Method for Integration

TL;DR

This paper introduces a unified substitution framework for integrating functions with quadratic radicals, simplifying the process by expressing complex exponentials in algebraic form and providing explicit templates that improve predictability and reduce complexity.

Contribution

The paper develops a branch-consistent method using explicit substitution templates derived from identities for exponential inverse trigonometric functions, unifying classical substitutions and enhancing computational efficiency.

Findings

Provides five explicit substitution templates for radicals

Recovers classical substitutions like Euler's and Weierstrass as special cases

Demonstrates improved predictability and reduced expression complexity in CAS benchmarks

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 $e^{\pm i\cos^{-1}(y)}$ 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.