Date: Aug 6, 2013 7:51 PM
Author: clicliclic@freenet.de
Subject: Rubi 4.1 and the Timofeev test suite


clicliclic@freenet.de schrieb:
>
> The following table summarizes the data made available for the example
> integrals in Timofeev's 1948 book. To simplify the accounting, the
> numbers of examples represent the actual numbers of distinct
> integrands; for chapters where this has not yet been determined, the
> total of Timofeev's numbered examples is given instead:
>
> ch. ex's Der. Fri. Mpl MMA Rubi etc. etc.
> 6.10 1.21 17 9 4.1
> -------------------------------------------------------
> 1 86 83 86 85 86 84
> 2 90 - - - - -
> 3 12 12 12 12 12 12
> 4 132 - - - - -
> 5 120 - - - - -
> 6 26 - - - - -
> 7 11 9 11 11 11 8
> 8 109 - - - - -
> 9 59 - - - - -
> -------------------------------------------------------
> 100% 95.4% 100% 99.1% 100% 95.4%
>
> The Winners so far are FriCAS 1.21 and Mathematica 9!
>
> It should be noted that some FriCAS results do not hold for all values
> of the integration variable (and/or parameters in the integrand).
>
> I will let matters rest at this point until new chapters or revised
> data become available.
>


I thought it would be instructive to list those integrals Rubi 4.1 fails
on:

Timofeev's example 29 from Chapter 1:

INT(TAN(x)*TAN(x - a), x) = 1/TAN(a)*LN(1 + TAN(a)*TAN(x)) - x

Timofeev's example 64 from Chapter 1:

INT(ASIN(x/a)^(3/2)/SQRT(a^2 - x^2), x)
= 2/5*ASIN(x/a)^(3/2)*ATAN(x/SQRT(a^2 - x^2))
= 2/5*(a/SQRT(a^2 - x^2))*SQRT(1 - (x/a)^2)*ASIN(x/a)^(5/2)

Timofeev's example 6 from Chapter 7:

INT(x*SIN(x)^3/COS(x)^4, x)
= x*(1/(3*COS(x)^3) - 1/COS(x)) - SIN(x)/(6*COS(x)^2)
+ 5/6*ATANH(SIN(x))

Timofeev's example 10 from Chapter 7:

INT((2*x + SIN(2*x))/(x*SIN(x) + COS(x))^2, x)
= - 2*COS(x)/(x*SIN(x) + COS(x))

Timofeev's example 11 from Chapter 7:

INT((x/(x*COS(x) - SIN(x)))^2, x)
= (x*SIN(x) + COS(x))/(x*COS(x) - SIN(x))

In my view, the first two failures are deplorable but understandable,
while the third one must be counted as a bug. The last two are simply to
be expected. Timofeev's book can be found at:

<http://www.math-life.com/jdownloads/view.download/27/279.html>

Martin.