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Re: Elementary vs non-elementary result.

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On 5/6/2015 11:52 AM, RGVickson@shaw.ca wrote:
> The following integration occurred in Physics Forums:
> int(f,x), where f = (x^2+1)/[(x^2-1)*sqrt(1+x^4)].
>
> If asked for the indefinite integral int(f,x), Maple 11 (and Maple 14) \
>return a nasty formula involving Elliptic functions. However, if asked
>for the definite integral int(f,x=0..z) assuming z>0, both versions of
>Maple return an explicit, closed-form function F(z)involving only
>elementary functions. Furthermore, dF/dz really does equal f(z)!
>(Initially, F(z) is complex because it involves an arctanh of an
>argument > 1, but an evalc on it yields a purely real expression
>defined for all z =/= 1, and again, which has the correct derivative.)
>
> I just wonder why Maple was able to find an elementary expression
>for the  definite integral (but at any variable upper limit) but not for the indefinite integral.
>
> RGV
>

Try Maple 2015.

restart;
f:=x-> (x^2+1)/((x^2-1)*sqrt(1+x^4)):
F:=int(f(x),x);

sqrt(1-I*x^2)*sqrt(1+I*x^2)*EllipticF(x*((1/2)*sqrt(2)+
(1/2*I)*sqrt(2)), I)/(((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*
sqrt(x^4+1))+2*(-1)^(3/4)*sqrt(1-I*x^2)*sqrt(1+I*x^2)*
EllipticPi((-1)^(1/4)*x, -I, sqrt(-I)/((-1)^(1/4)))/sqrt(x^4+1)

expand(simplify(diff(F,x)));
x^2/((x^2-1)*sqrt(x^4+1))+1/((x^2-1)*sqrt(x^4+1))

------------------------------

F:=int(f(x),x=0..z) assuming z>0;
  (1/2-(1/2)*I)*(1-I*z^2)^(1/2)*(1+I*z^2)^(1/2)*(-(-1)^(1/4)*
EllipticPi((-1)^(1/4)*z, -I, I)*2^(1/2)+(-1)^(3/4)*
EllipticPi((-1)^(1/4)*z, -I, I)*2^(1/2)+EllipticF((1/2)*2^(1/2)*z
+((1/2)*I)*z*2^(1/2), I))*2^(1/2)/(z^4+1)^(1/2)

expand(simplify(diff(F,z)));
z^2/((z^2-1)*sqrt(z^4+1))+1/((z^2-1)*sqrt(z^4+1))


--Nasser

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Thread

Elementary vs non-elementary result. RGVickson@shaw.ca - 2015-05-06 09:52 -0700
  Re: Elementary vs non-elementary result. "Nasser M. Abbasi" <nma@12000.org> - 2015-05-06 15:41 -0500
    Re: Elementary vs non-elementary result. RGVickson@shaw.ca - 2015-05-06 16:09 -0700
      Re: Elementary vs non-elementary result. "Nasser M. Abbasi" <nma@12000.org> - 2015-05-07 01:50 -0500
        Re: Elementary vs non-elementary result. acer <maple@rogers.com> - 2015-05-08 10:28 -0700
          Re: Elementary vs non-elementary result. "G. A. Edgar" <edgar@math.ohio-state.edu.invalid> - 2015-05-09 07:44 -0600
            Re: Elementary vs non-elementary result. acer <maple@rogers.com> - 2015-05-11 20:16 -0700
  Re: Elementary vs non-elementary result. acer <maple@rogers.com> - 2015-05-08 09:59 -0700
    Re: Elementary vs non-elementary result. RGVickson@shaw.ca - 2015-05-08 15:03 -0700

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