A118178 integral in closed form

[David W. Cantrell]

----- Original Message ----- 
From: "Eric W. Weisstein" <eric at weisstein.com>
To: "Sequence Fans Mailing List" <seqfan at ext.jussieu.fr>
Sent: Wednesday, April 19, 2006 00:29
Subject: A118178 integral in closed form

> Can anyone get
>
> http://www.research.att.com/~njas/sequences/A118178
>
> 4*Integrate[Sqrt[4*Sin[t]^4-5*Sin[t]^2+2], {t, 0, Pi/2}]
>
> in closed form?

It can, of course, be expressed in terms of complete elliptic
integrals.

Using Mathematica notation, with m = (4 + Sqrt[2])/8, it's

4*2^(1/4)*(EllipticE[m] - EllipticK[m])
+ (3 + 2*Sqrt[2])*2^(-1/4)*EllipticPi[(4 - 3*Sqrt[2])/8, m]

It disturbs me that Mathematica cannot get this (and many similar
results). I took me quite a while to get the result by hand. But the
process is straightforward. If it were implemented in a CAS, getting
the result above should have taken almost no time. Can Maple or some
other CAS get the above result (or something equivalent)?

BTW, in <http://mathworld.wolfram.com/EightCurve.html>, you note that

"The arc length can be computed in (complicated) closed form for
0 < t < 1.33, but hits a branch cut jump near that value."

I know exactly what you're talking about. Before I did it by hand, I
tried to use Mathematica. Version 5.1 gave me an antiderivative, but
it was inadequate for the task at hand due to a jump discontinuity at
roughly 4/3. There is absolutely no reason to have such a
discontinuity! The integrand is continuous on R, and so there is an
antiderivative valid on R. And it happens in this case that such an
antiderivative can be expressed in terms of elliptic integrals. I'm
very sorry to see that Mathematica can't do so.

Regards,
David