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.
>
> RJF
>
>>
>>
--
Michel Talon