
Twoterm trinomal integration recurrences found
Posted:
Jun 30, 2011 4:33 AM


Section 2.160 of Gradshteyn & Ryzhik?s ?Table of Integrals, Series, and Products? gives formulas for integrating expressions of the form x^m (a+b x^k+c x^(2k))^n wrt x. Unfortunately rules based on these formulas are horribly inefficient due to recursive fanout. For example, Rubi 2 required 232 steps to integrate x^10/sqrt(a+b x+c x^2).
In the process of resolving this problem for Rubi 3, I discovered (rediscovered?) some amazing twoterm recurrence equations for integrating such expressions. Rules based on these recurrences do not suffer from recursive fanout, nor do they require partial fraction expansion or advanced methods like the Risch algorithm. Now it takes Rubi 3 only 11 steps to integrate the above expression.
Achieving twoterm closure for the recurrences required generalizing the integrands to expressions of the form x^m (A+B x^k) (a+b x^k+c x^(2k))^n. The 6 recurrences are listed in the pdf file
http://www.apmaths.uwo.ca/RuleBasedMathematics/TrinomialRecurrenceEquations.pdf
(A Mathematica .nb notebook file is also available.) The following summarizes how the recurrences transform the exponents {m,n} of the integrand:
Recurrence 1: {m,n} > {mk,n+1} Recurrence 2: {m,n} > {mk,n} Recurrence 3: {m,n} > {m,n1} Recurrence 4: {m,n} > {m+k,n1} Recurrence 5: {m,n} > {m+k,n} Recurrence 6: {m,n} > {m,n+1}
Note that the form of these 6 trinomial recurrences and their effect on exponents are strikingly similar to the 6 recurrences for integrands of the form sin(x)^m (A+B sin(x)^k) (a+b sin(x)^k)^n when a^2=b^2 discussed at length in the sci.math.symbolic thread ?Trig integration rules sought?. There must be some grand underlying symmetry responsible for this similarity. Perhaps someone smarter than me can figure it out...
Albert

