Richard Fateman wrote:
> integrate(2^x/sqrt(1+4^x), x);
> is not done.
>
> but integrate(radcan (2^x/sqrt(1+4^x)), x);
> works fine.
>
> why this is odd? the risch algorithm should be decomposing stuff using
> radcan.
> It has to notice that 4^x is 2 ^(2*x) , and is therefore related to 2^x.
>
> Macsyma has the same problem and the same fix.
>
> This problem was posed as something the new "Fricas" can do.
> RJF
At first i had read x^4! I would be happier if maxima could integrate
this one than a completely exotic and artificial integral with 4^x.
Anyways, Maple, Mathematica have no problem integrating elliptic
differentials, and this is not a super sophisticated risch algorithm
for algebraic integrands, it is simply recognizing that integrals of the
form integrate(r(x)*sqrt(P(x)),x) where r(x) is a rational function and
P is a polynomial of degree 3 or 4 are elliptic,
the explicit elliptic form being obtained by writing the curve y^2=P(x) in
canonical elliptic form through reparametrizations. More complicated
algebraic integrals cannot be expressed on well known special functions,so
it is useless to develop an algorithm for more general cases.
--
Michel Talon