Michel Talon wrote:
> Richard Fateman wrote:
>
>
>> 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}
>>
>
> This is not more general that what i advocated, since R(x,q) can be
> written (a(x)+b(x)q)/(c(x)+d(x)q) since q^2 is a polynomial, and then
> multiplying numerator and denominator by (c-dq) you get r1(x)+qr2(x).
> Only the second term is non trivial. I reiterate that the general algebraic
> equation P(x,q)=0 is totally useless, since the curve will be hyperelliptic
> or worse, that is the integral cannot be expressed by known special
> functions. The only case of interest boils down to q^2= P(x) with P a
> polynomial of degree 3 or 4. This can always been brought to the form
> P = x^3 + px + q by an homography bringing one of the roots of P at
> infinity, and a translation to kill the x^2 term. This is a canonical form
> for elliptic curves. I don't know what is this stuff about Carlson method,
> what i sketched above is described in all books on Riemann surfaces or
> whatever, for example Springer Introduction to Riemann surfaces, i am
> quite sure it is even in Whittaker and Watson.
>
FWIW, I tried implementing the reduction algorithm in A&S for converting
R(x,q) to elliptic integals of the first, second, and third kind. Parts
of it work, but many parts do not. In particular, I never got it to
recognize that sqrt((1-x^2)/(1-m*x^2)) is an elliptic integral.
Ray