       Re: Try it again, Mathematica! Was: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))

• To: mathgroup at smc.vnet.net
• Subject: [mg90840] Re: Try it again, Mathematica! Was: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Sat, 26 Jul 2008 04:19:53 -0400 (EDT)
• References: <g6710s\$sb6\$1@smc.vnet.net> <g6c959\$bpg\$1@smc.vnet.net>

Alois Steindl <Alois.Steindl at tuwien.ac.at> wrote:
> Hello,
> I played around a little bit and found the following interesting
> behaviour:
>
> In:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
> Out:= \!\(
> \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Pi]\)]\(
> SqrtBox[\(5 - 4\ Cos[t] + Cos[2\ t]\)] \[DifferentialD]t\)\)
>
> In:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}]
> Out:= 1/5 Sqrt[
>  2 + 4 I] (-5 I EllipticE[-(3/5) - (4 I)/5] + (2 + I) Sqrt
>      EllipticE[-(3/5) + (4 I)/5] - (12 - 4 I) EllipticK[-(3/5) - (
>       4 I)/5] + (6 - 2 I) Sqrt EllipticK[8/5 - (4 I)/5] +
>    4 I Sqrt
>      EllipticPi[
>      1/5 + (2 I)/5, -(3/5) + (4 I)/5] + (8 + 4 I) EllipticPi[
>      1 - 2 I, -(3/5) - (4 I)/5])
>
> So it seems, it helps to ask Mathematica twice.

On my system, one must ask Mathematica _three_ times. (The third time's the
charm.)

The reason Mathematica eventually succeeded is that it remembered work from
its previous attempts. If we had done ClearSystemCache[] between attempts,
Mathematica would presumably never have succeeded.

Is there a parameter which we can change so that Mathematica succeeds on
the first try?

David

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