Vladimir Bondarenko <vb@[EMAIL PROTECTED]
> writes:
> Is there an Insensible-To-Fear Human Warrior the Simplifier to
> come up with a succession of CAS commands to squeeze this very,
> very, very much
Insensible? Yup, that part sounds like me...
> ((-1)^(1/4)*((1-I)*Pi+I*(Pi-ArcTan[4])+I*ArcTan[4]+(2*I)*ArcTan
> [2+(1-I)*Sqrt[2]]-2*ArcTan[2+(1+I)*Sqrt[2]]+I*ArcTan[(-6+2*Sqrt
> [2])/(-1+2*Sqrt[2])] - I*ArcTan[(2+2*Sqrt[2])/(1+2*Sqrt[2])]-I*
> ArcTan[(5+2*Sqrt[2])/(4+2*Sqrt[2])]+I*ArcTan[(5+4*Sqrt[2])/(4+4
> *Sqrt[2])] +I*ArcTan[(6+3*Sqrt[3])/(6+2*Sqrt[3])]-I*ArcTan[(69+
> 12*Sqrt[3])/(114 -22*Sqrt[3])]+2*ArcTanh[(14-4*Sqrt[3])/(20-11*
> Sqrt[3])] + (2*I)*ArcTanh[((6+4*I)+2*Sqrt[3])/((3+4*I)-(1+2*I)*
> Sqrt[3])]+(1+I)*Log[-6+7*Sqrt[3]]-I*((-2*I)*Pi+I*(Pi-ArcCot[4])
> +I*(Pi+ArcTan[(-1+2*Sqrt[2])/(-6+2*Sqrt[2])])-Log[17]/2+Log[(-6
> +2*Sqrt[2])^2 +(-1+2*Sqrt[2])^2]/2)+Log[(-6+2*Sqrt[2])^2+(-1+2*
> Sqrt[2])^2]/2 + I*( (-I)*ArcCot[4]+I*ArcTan[(1+2*Sqrt[2])/(2+2*
> Sqrt[2])]-Log[17]/2+Log[(1+2*Sqrt[2])^2+(2+2*Sqrt[2])^2]/2)-Log
> [(1+2*Sqrt[2])^2+(2+2*Sqrt[2])^2]/2+I*(I*ArcTan[(4+2*Sqrt[2])/(
> 5+2*Sqrt[2])]+Log[(4+2*Sqrt[2])^2+(5+2*Sqrt[2])^2]/2)-Log[(4+2*
> Sqrt[2])^2 + (5+2*Sqrt[2])^2]/2-I*(I*ArcTan[(4+4*Sqrt[2])/(5+4*
> Sqrt[2])] + Log[(4 + 4*Sqrt[2])^2+(5+4*Sqrt[2])^2]/2)+Log[(4+4*
> Sqrt[2])^2 + (5+4*Sqrt[2])^2]/2+I*(I*ArcTan[(6+2*Sqrt[3])/(6+3*
> Sqrt[3])]+Log[(6+2*Sqrt[3])^2+(6+3*Sqrt[3])^2]/2)+Log[(6+2*Sqrt
> [3])^2 +(6+3*Sqrt[3])^2]/2-I*(I*ArcTan[(114-22*Sqrt[3])/(69+12*
> Sqrt[3])]+Log[(114-22*Sqrt[3])^2+(69+12*Sqrt[3])^2]/2)-Log[(114
> -22*Sqrt[3])^2+(69+12*Sqrt[3])^2]/2))/4
Maple 11:
> Q:= MmaTranslator[FromMma](" ...that awful mess... "):
Q1:= simplify(combine(simplify(convert(Q,ln))));
Q1 := -1/8*2^(1/2)*(arctan(56/17*2^(1/2))-2*Pi)
Squeezed enough? The following looks like it might be interesting:
> cos(4*sqrt(2)*Q1);
17/81
So what would be cos(sqrt(2)*Q1)?
> solve(orthopoly[T](4,x) = %, x);
1/3, -1/3, 2/3*2^(1/2), -2/3*2^(1/2)
One of those... and evalf confirms that it's 1/3.
And so the answer is:
arccos(1/3)/sqrt(2);
--
Robert Israel israel@[EMAIL PROTECTED]
of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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