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Topic: The Charlwood Fifty
Replies: 52   Last Post: Jun 24, 2013 10:24 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: The Charlwood Fifty
Posted: May 23, 2013 12:17 PM

Albert Rich schrieb:
>
> There has been much discussion in recent sci.math.symbolic posts about
> the performance of various systems on 10 integration problems from
> Kevin Charlwood's 2008 article "Integration on Computer Algebra
> Systems". An appendix to his article includes 40 more problems.
>
> It seems to me if these problems are going be used as a test-suite, we
> should start by trying to reach consensus as to what the best answer
> is. To that end I have posted a pdf file at
>
> http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf
>
> showing the 50 integrals and the best antiderivatives I have found so
> far. This was done with the help of the article as well as
> Mathematica, Maple, Rubi and Derive.
>
> If you should find substantially better ones and would like to improve
> the test-suite, please post them on sci.math.symbolic so I can include
> them in the Charlwood Fifty. Substantial improvements include
> significantly simpler and more compact, involve elementary rather than
> special functions, involve special rather than hypergeometric
> functions, real rather than complex, continuous rather than
> discontinuous, etc.
>
> Albert

Hello,

here are my antiderivatives for problems #1 to #10, as promised. I think
they are as compact, continuous, real, and elementary as one could wish
them to be:

INT(ASIN(x)*LN(x), x) =
= x*ASIN(x)*(LN(x) - 1) + LN(SQRT(1 - x^2) + 1)
+ (SQRT(1 - x^2) - 1)*LN(x) - 2*SQRT(1 - x^2)

INT(x*ASIN(x)/SQRT(1 - x^2), x) =
= x - SQRT(1 - x^2)*ASIN(x)

INT(ASIN(SQRT(x + 1) - SQRT(x)), x) =
= (x + 3/8)*ASIN(SQRT(x + 1) - SQRT(x)) + SQRT(2)/8
*(3*SQRT(x + 1) + SQRT(x))*SQRT(SQRT(x)*(SQRT(x + 1) - SQRT(x)))

INT(LN(1 + x*SQRT(1 + x^2)), x) =
= SQRT(2*SQRT(5) + 2)/2*ATAN(SQRT(x^2 + 1)*SQRT(2*SQRT(5) + 2)/2)
+ SQRT(2*SQRT(5) + 2)/2*ATAN(x*SQRT(2*SQRT(5) - 2)/2)
+ SQRT(2*SQRT(5) - 2)/2*LN(SQRT(2)*x + SQRT(SQRT(5) - 1))
- SQRT(2*SQRT(5) - 2)/2*LN(SQRT(2)*SQRT(x^2 + 1) + SQRT(SQRT(5) + 1))
+ x*LN(x*SQRT(x^2 + 1) + 1) - 2*x

INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
= x/3 + 1/3*ATAN(SIN(x)*COS(x)*(COS(x)^2 + 1)
/(COS(x)^2*SQRT(COS(x)^4*COS(x)^2 + 1) + 1))

INT(TAN(x)*SQRT(1 + TAN(x)^4), x) =
= 1/2*(SQRT(1 + TAN(x)^4) - ASINH(TAN(x)^2)
- SQRT(2)*ATANH((1 - TAN(x)^2)/(SQRT(2)*SQRT(1 + TAN(x)^4))))

INT(TAN(x)/SQRT(SEC(x)^3 + 1), x) =
= - 2/3*ATANH(SQRT(1 + SEC(x)^3))

INT(SQRT(TAN(x)^2 + 2*TAN(x) + 2), x) =
= ASINH(TAN(x) + 1) + SQRT(2*SQRT(5) + 2)/2
*ATAN(SQRT(2)*(TAN(x)*SQRT(SQRT(5) + 1) - SQRT(SQRT(5) - 1))
/(2*SQRT(TAN(x)^2 + 2*TAN(x) + 2))) - SQRT(2*SQRT(5) - 2)/2
*ATANH(SQRT(2)*(TAN(x)*SQRT(SQRT(5) - 1) + SQRT(SQRT(5) + 1))
/(2*SQRT(TAN(x)^2 + 2*TAN(x) + 2)))

INT(SIN(x)*ATAN(SQRT(SEC(x) - 1)), x) =
= COS(x)*(1/2*SQRT(SEC(x) - 1) - ATAN(SQRT(SEC(x) - 1)))
- 1/2*ATAN(1/SQRT(SEC(x) - 1))

INT(x^3*EXP(ASIN(x))/SQRT(1 - x^2), x) =
= EXP(ASIN(x))/10*(x^3 + 3*x - 3*(x^2 + 1)*SQRT(1 - x^2))

Many of them agree (or almost agree) with your solutions. The result for
#5 can be shortened somewhat if continuity is of no concern:

INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
= - 1/3*ATAN(COT(x)*COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1))

I have just the time to add a few observations on your solutions for
Charlwood's integrals #11 to #50: The integration variable for problems
#27, #30, #31, #32, #33, #34, #35 is theta, not x. Problems #28 and #29
are (of course) pseudo-elliptic:

INT((1 + x^2)/((1 - x^2)*SQRT(1 + x^4)), x) =
= 1/SQRT(2)*ATANH(SQRT(2)*x/SQRT(1 + x^4))

INT((1 - x^2)/((1 + x^2)*SQRT(1 + x^4)), x) =
= 1/SQRT(2)*ATAN(SQRT(2)*x/SQRT(1 + x^4))

When these two integrands and their antiderivatives are averaged, one
obtains a pseudo-elliptic integral already known to Euler:

INT(SQRT(1 + x^4)/(1 - x^4), x) =
= SQRT(2)/4*ATAN(SQRT(2)*x/SQRT(1 + x^4))
+ SQRT(2)/4*ATANH(SQRT(2)*x/SQRT(1 + x^4))

See the postscript to the Lettre de Fuss à Condorcet of May 1778,
reprinted in the Bulletin des Sciences Mathématiques et Astronomiques
(2e série), tome 3, no 1 (1879), p. 225-227:

<http://www.numdam.org/item?id=BSMA_1879_2_3_1_225_0>

Martin.