Two-term recurrence formulae for indefinite algebraic integrals

TL;DR

This paper presents 136 two-term recurrence relations for indefinite integrals involving algebraic and exponential functions, enabling systematic computation by adjusting polynomial exponents without changing integrand form.

Contribution

It introduces a comprehensive set of recurrence relations for a wide class of algebraic and exponential integrals, expanding computational tools for symbolic integration.

Findings

136 recurrence relations derived

Relations preserve integrand form while changing exponents

Facilitates systematic integral computation

Abstract

Two-term recurrence relations are supplied for indefinite integrals of functions that involve factors of the types ${P_2}^n$, ${P_3}^n$, ${P_4}^n$, ${P_1}^m {Q_1}^n$, $E_1 {P_1}^n$, ${P_1}^m {Q_2}^n$, $E_1 {P_2}^n$, ${P_2}^m {Q_2}^n$, ${P_1}^m {Q_1}^n {S_1}^p$, $E_1 {P_1}^m {Q_1}^n$, ${P_1}^m {Q_1}^n {S_2}^p$, and ${P_1}^m {Q_1}^n {S_1}^p {T_1}^q$, where $P_i$, $Q_j$, $S_k$ and $T_l$ denote arbitrary polynomials of degree $i$, $j$, $k$ and $l$ in the integration variable, $E_1$ represents the exponential function of an arbitrary linear polynomial in this variable, and $m$, $n$, $p$ and $q$ are arbitrary constant exponents. The 136 relations leave the form of an integrand unchanged and increment or decrement the exponents in steps of unity.