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Two-term trinomal integration recurrences found
Posted:
Jun 30, 2011 4:33 AM
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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 fan-out. 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 two-term recurrence equations for
integrating such expressions. Rules based on these recurrences do not
suffer from recursive fan-out, 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 two-term 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} --> {m-k,n+1}
Recurrence 2: {m,n} --> {m-k,n}
Recurrence 3: {m,n} --> {m,n-1}
Recurrence 4: {m,n} --> {m+k,n-1}
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
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