Michel Talon wrote:
> 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.
>
I agree.
> 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.
>
The approach in the literature is to attack the problem of
integrate(R(x,q),x) where R is a rational function, and q satisfies
an algebraic equation in x, e.g. q^2= x^4+1. There are several
approaches. For one reference, try google {ng elliptic macsyma}
RJF
>
>