Two-term recurrence formulae for indefinite algebraic integrals

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.