Date: Jul 14, 2013 1:25 AM
Author: daly@axiom-developer.org
Subject: Re: An independent integration test suite

On Sunday, July 14, 2013 12:22:18 AM UTC-4, da...@axiom-developer.org wrote:
> On Sunday, February 24, 2013 10:11:47 AM UTC-5, clicl...@freenet.de wrote:
>

> > The Russian book "Integration of Functions"
>
> >
>
> > (?????????????? ???????) published by A.F. Timofeev
>
> >
>
> > (?.?. ????????) in 1948 provides many integration Examples
>
> >
>
> > (???????) of the following general kind:
>
> >
>
> >
>
> >
>
> > INT(LN(x)^2*x, x) = x^2/2*(LN(x)^2 - LN(x) + 1/2)
>
> >
>
> >
>
> >
>
> > INT(LN(x)/x^5, x) = - (4*LN(x) + 1)/(16*x^4)
>
> >
>
> >
>
> >
>
> > INT(x^2*LN((x - 1)/x), x) = x^3/3*LN((x - 1)/x) - LN(x - 1)/3 - x*(x +
>
> >
>
> > 2)/6
>
> >
>
> >
>
> >
>
> > INT(COS(x)^5, x) = SIN(x)/15*(3*COS(x)^4 + 4*COS(x)^2 + 8)
>
> >
>
> >
>
> >
>
> > INT(SIN(x)^2*COS(x)^4, x) = 1/6*SIN(x)^3*COS(x)^3 + 1/8*SIN(x)^3*COS(x)
>
> >
>
> > - 1/16*SIN(x)*COS(x) + x/16
>
> >
>
> >
>
> >
>
> > INT(1/SIN(x)^5, x) = 3/8*LN(TAN(x/2)) - 3*COS(x)/(8*SIN(x)^2) -
>
> >
>
> > COS(x)/(4*SIN(x)^4)
>
> >
>
> >
>
> >
>
> > INT(SIN(x)/#e^x, x) = - (COS(x) + SIN(x))/(2*#e^x)
>
> >
>
> >
>
> >
>
> > INT(#e^(2*x)*SIN(3*x), x) = 1/13*#e^(2*x)*(2*SIN(3*x) - 3*COS(3*x))
>
> >
>
> >
>
> >
>
> > INT(a^x*COS(x), x) = a^x/(LN(a)^2 + 1)*(LN(a)*COS(x) + SIN(x))
>
> >
>
> >
>
> >
>
> > INT(COS(LN(x)), x) = x/2*(COS(LN(x)) + SIN(LN(x)))
>
> >
>
> >
>
> >
>
> > INT(SEC(x)^2*LN(COS(x)), x) = TAN(x)*LN(COS(x)) + TAN(x) - x
>
> >
>
> >
>
> >
>
> > INT(x*TAN(x)^2, x) = LN(COS(x)) + x*TAN(x) - 1/2*x^2
>
> >
>
> >
>
> >
>
> > INT(ASIN(x)/x^2, x) = - ASIN(x)/x + LN((1 - SQRT(1 - x^2))/x)
>
> >
>
> >
>
> >
>
> > INT(ASIN(x)^2, x) = x*ASIN(x)^2 + 2*SQRT(1 - x^2)*ASIN(x) - 2*x
>
> >
>
> >
>
> >
>
> > INT(x^2*ATAN(x)/(1 + x^2), x) = x*ATAN(x) - 1/2*ATAN(x)^2 - 1/2*LN(x^2 +
>
> >
>
> > 1)
>
> >
>
> >
>
> >
>
> > INT(ACOS(SQRT(x/(x + 1))), x) = (x + 1)*(ACOS(SQRT(x/(x + 1))) +
>
> >
>
> > SQRT(1/(x + 1))*SQRT(x/(x + 1)))
>
> >
>
> >
>
> >
>
> > There are 8 chapters in the book containing 81+90+14+132+120+26+11+59 =
>
> >
>
> > 533 integration examples; the above are numbers 76 to 81 from chapter 1.
>
> >
>
> > I believe the whole would make a good independent integration test suite
>
> >
>
> > because the book antedates all automated integrators and because the
>
> >
>
> > author claims to cover the field thoroughly (with respect to integrals
>
> >
>
> > expressible by elementary functions and by elliptic integrals). In the
>
> >
>
> > Foreword (???????????), he writes (as translated by Google):
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> >
>
> >
>
> >
>
> > "In most of the guides on higher mathematics the question of integration
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> >
>
> > of functions of one independent variable has a fairly complete coverage,
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> >
>
> > so that very often students do not get a clear idea of ??what functions
>
> >
>
> > are integrated in closed form, for which this integration is impossible,
>
> >
>
> > and what techniques are useful in a particular case for various kinds of
>
> >
>
> > functions. With this in mind, the author of this book sought to explain
>
> >
>
> > the issue to a possible full, paying particular attention to the
>
> >
>
> > practice of integration, thus introducing a large number of examples.
>
> >
>
> > Thus, this book can serve as a first, a reference book for those wanting
>
> >
>
> > to get a quick response with respect to a quadrature, and secondly, aid
>
> >
>
> > for students who wish to supplement and deepen their knowledge in this
>
> >
>
> > matter."
>
> >
>
> >
>
> >
>
> > Some may prefer the original:
>
> >
>
> >
>
> >
>
> > "? ??????? ????? ?????????? ?? ?????? ?
>
> >
>
> > ????????? ?????? ?? ??????????????
>
> >
>
> > ??????? ?????? ???????????? ?????????
>
> >
>
> > ?? ?? ????? ?????????? ???????
>
> >
>
> > ?????????, ?????????? ???? ????? ?????
>
> >
>
> > ???????? ?? ???????? ??????
>
> >
>
> > ????????????? ? ???, ????? ??????? ????
>
> >
>
> > ????????? ? ???????? ????, ???
>
> >
>
> > ????? ??? ?????????????? ?????????? ?
>
> >
>
> > ????? ?????? ?????????????
>
> >
>
> > ????????? ? ??? ??? ???? ?????? ??? ???
>
> >
>
> > ?????? ????? ???????. ???? ??? ?
>
> >
>
> > ????, ????? ? ????????? ????? ?????????
>
> >
>
> > ???????? ?????? ? ?????????
>
> >
>
> > ????????, ??????? ?????? ???????? ?? ??
>
> >
>
> > ?????? ??????????????, ????? ???
>
> >
>
> > ???? ??????? ?????????? ????????. ????
>
> >
>
> > ? ???????, ????? ??? ?????
>
> >
>
> > ???????, ??-??????, ???????????? ??? ??
>
> >
>
> > ?, ???????? ???????? ?????? ?????
>
> >
>
> > ???????????? ??? ??? ???? ??????????, ?
>
> >
>
> > ??-??????, ???????? ???
>
> >
>
> > ????????, ???????? ????????? ? ???????
>
> >
>
> > ? ???? ?????? ? ???? ???????."
>
> >
>
> >
>
> >
>
> > A djvu file of the book (6 Megabyte) can be downloaded freely from
>
> >
>
> > various websites - just google the Russian title. The full name of the
>
> >
>
> > author apparently was Aleksandr Fedotovich Timofeev (?????????
>
> >
>
> > ????????? ????????), but I couldn't locate any
>
> >
>
> > information on him beyond this. A Russian review of the book is
>
> >
>
> > available at:
>
> >
>
> >
>
> >
>
> > <http://www.mathnet.ru/links/5566c95518efdede979ffdc8a2278b01/rm8600.pdf>
>
> >
>
> >
>
> >
>
> > There are the usual misprints in the book, but with both the integrand
>
> >
>
> > and antiderivative available, the original meaning can be reconstructed
>
> >
>
> > for all (or almost all) examples I think. Timofeev's antiderivatives are
>
> >
>
> > usually close to the most compact form possible (an exception is his
>
> >
>
> > consistent use of logarithms for inverse hyperbolic functions), but they
>
> >
>
> > have to be checked for validity over the complex plane, and be corrected
>
> >
>
> > if necessary (this seems rare, but was needed for the last example
>
> >
>
> > above). Apart from compactness, continuity (and reality) on the real
>
> >
>
> > axis might be worth checking and repairing too.
>
> >
>
> >
>
> >
>
> > So, if 5 to 10 people were found willing to type in and check (and
>
> >
>
> > perhaps correct or improve) 50 to 100 integrals and evaluations each, a
>
> >
>
> > digitization of this corpus could be an almost pleasant task, and surely
>
> >
>
> > quite useful. What do you think?
>
> >
>
> >
>
> >
>
> > Martin.
>
>
>
> Axiom has published a Computer Algebra Test Suite at
>
> http://axiom-developer.org/axiom-website/CATS/index.html
>
>
>
> It includes Schaums integrals and Kamke's Ordinary Differential Equations.
>
> It also includes Albert Rich's integration set.
>
> In all there are several thousand examples.
>
>
>
> The source file format is latex, the output file format is pdf.
>
> The axiom.sty package is at
>
> http://axiom-developer.org/axiom-website/CATS/axiom.sty
>
>
>
> Each problem includes the source input.
>
> Axiom's output is prefixed with --R which is an Axiom comment.


The file format is in groups of 5 expressions:
tnnn:= the input expression
rnnn:= the expected, published result
annn:= integration of tnnn (Axiom's result)
mnnn:= difference of annn and rnnn (should be constant, hopefully 0)
dnnn:= derivative of mnnn (to see if the difference is constant)

There are, as usual, a LOT of mathematical and symbolic issues
such as branch cuts and simplification. If I remember correctly,
Axiom and Mathematics share one set of cuts, Maple and Maxima
share a different set.

The schaums suite has attempts to overcome the simplification
issues by using various Axiom routines. Often a special routine
can discover that a result is actually correct but different by
a constant. It is also a useful way to discover errors in the
original published source.

I have not yet done this with the Rich test suite due to the
size of the problem and the lack of time.

I will look at the Charlwood test suite.

It has been a long-standing Axiom project goal to develop CATS,
a Computer Algebra Test Suite.

I'm open to suggestion about other sources.

I believe we need a system-independent source of algorithms
that will perform these integrations. I'd like to see NIST
or some other standard organization support such an effort.
It is not enough to just publish the equations with answers.
We need to have excellent "standard" algorithms.

It would also be useful to standardize on a system-independent
machine-readable notation that had published conversion routines
so we could all use the same input for equations and results.
It took a while to hand-write the several thousand equations.

We moved past tables of logarithms a few years ago.
It is time to move past tables of equations.
This is, after all, mathematics not art.