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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: Jun 17, 2013 12:11 PM

"Nasser M. Abbasi" schrieb:
>
> On 6/16/2013 11:30 AM, clicliclic@freenet.de wrote:

> >
>
> >
> > Oops, this should have read:
> >
> > 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))
> >

>
> fyi, for the above, which #5, the LeafCount is 45
>
> ---------------------------------------
>
> f1= x/3+1/3*ArcTan[Sin[x]*Cos[x]*(Cos[x]^2+1)/(Cos[x]^2*Sqrt[Cos[x]^4+Cos[x]^2+1]+1)];
>
> LeafCount[f1]
> -----------------------------------------
> 45
>
> For the one listed in the table, it is 37:
>
> ------------------------------------------
>
> f0 =(-ArcSin[Cos[x]^3]*Sqrt[1-Cos[x]^6]*Csc[x])/(3*Sqrt[1+Cos[x]^2+Cos[x]^4]);
>
> LeafCount[f0]
>
> Out[13]= 37
> ---------------------------------------
>
> I was wondering: assuming both antiderivates contain
> just elemetary functions, can one then use the leaf count as a
> measure of which is most optimal answer? Or will there
> be other considerations one should look at or better
> way to measure which one is more "optimal" than the
> other?
>
> reference:
> http://reference.wolfram.com/mathematica/ref/LeafCount.html
>

Apart from the compactness of antiderivatives, as measured by leaf
counting, continuity on the real axis and absence of complex
intermediate results when evaluated on the real axis (which implies
absence of the imaginary unit) are important in my view, and usually
take precedence over compactness.

Thus, my 45-leafed result is fully continuous along the real axis,
whereas the shorter ATAN alternative:

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))

as well as Albert's 37-leafed ASIN version:

INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
= - ASIN(COS(x)^3)*SQRT(1 - COS(x)^6)*CSC(x)
/(3*SQRT(1 + COS(x)^2 + COS(x)^4))

jump at x = -pi, 0, pi, etc. This constitutes an unnecessary obstacle in
definite integration - imagine some quantity integrated along the path
of an orbiting spacecraft.

I usually accept logarithmic evaluations like INT(1/x, x) = LN(x), which
can be complex where the integrand is real (here for x < 0). I think
that users (e.g. calculus students) who need this integral from x = -2
to x = -1, say, should be able to accept that constants involving some
formal quantity #i appear which drop out of the final result.

Martin.