Integration

are you sure for your result?

Comment on 2007-10-17 16:04 from Axel Vogt ( Maple Rating 4

994)

  Int(exp(arcsin(x)), x);
  Change(%,arcsin(x)=y,y);
  value(%): simplify(%);

                           /
                          |
                          |  exp(y) cos(y) dy
                          |
                         /


                     1/2 exp(y) (cos(y) + sin(y))

Now check for a definite integral, say over [-1, +1] and [+1, +2] ...

with(IntegrationTools)

Comment on 2007-10-17 16:50 from jpmay ( Maple Rating 2

170)

You'll want to do a: with(IntegrationTools); before that or IntegrationTools[Change] of course.

That is correct

Comment on 2007-10-17 16:58 from harvam55 (2)

That is correct, but you are telling maple what subsitution to use, which in the case of this integral is the hardest part (I think). I was more interested in why maple does not know how to do this on it's own, not the solution. I guess I should have made that clear in the question.

int(exp(arcsin(x)),x)

Comment on 2007-10-17 16:42 from DJKeenan ( Maple Rating 2

191)

int(exp(arcsin(x)),x);

int(exp(arcsin(x)),x)

So Maple does not integrate it.

I would be interested in an explanation. My (admittedly very poor) understanding is that (i) Maple uses the Risch algorithm for integrating, (ii) this algorithm can integrate any elementary function, and (iii) exp(arcsin(x)) is an elementary function—and thus exp(arcsin(x)) should be integrated by Maple.

Why?

Comment on 2007-10-17 16:50 from harvam55 (2)

I am also insterested in why maple cannot do it without the user specifying what subsitution to use.

No complete implementation

Comment on 2007-10-17 18:56 from Robert Israel ( Maple Rating 4

1555)

I think what you mean is that it can integrate any function that has an elementary antiderivative. It certainly can't integrate elementary functions whose antiderivatives are not elementary. This one, though, does have an elementary antiderivative. It is an example of the mixed transcendental-algebraic case, because arcsin(x) = -i ln(sqrt(1-x^2) + i x). That is not completely implemented in Maple (or in any other CAS, apparently). Even the purely algebraic case is not completely implemented: see e.g.
this article by Manuel Bronstein in comp.soft-sys.math.maple from November 2000. I think what he says there is still true today. Note that Bronstein's example

> int(x / sqrt(x^4 + 10*x^2 - 96*x - 71), x);

is still not handled by Maple: it returns a non-elementary antiderivative involving elliptic functions, rather than the elementary antiderivative
-log((x^6+15*x^4-80*x^3+27*x^2-528*x+781) *<br />
sqrt(x^4+10*x^2-96*x-71)<br />
- x^8 - 20*x^6 + 128*x^5 - 54*x^4 + 1408*x^3 - 3124*x^2 - 10001)/8

integration algorithms

Comment on 2007-10-18 00:40 from DJKeenan ( Maple Rating 2

191)

Er, yup, I meant any elementary function that has an elementary antiderivative.

As for Mathematica, I just tried the example you gave on Mathematica's online Integrator, and that too returned something involving elliptic functions. I do not see this as proof that Mathematica does not have the full Risch algorithm though. Couldn't it just mean that the heuristic approaches took priority, and since they got an answer, nothing further was tried? Mathematica's Integrator can integrate exp(arcsin(x)).

As for other systems, I asked Joel Moses a long time ago, and he said that Macsyma did have the algorithm fully implemented. (Moses was responsible for Macsyma's symbolic integration routines.)

Macsyma

Comment on 2007-10-18 19:11 from Robert Israel ( Maple Rating 4

1555)

I suspect he was talking about the transcendental case.
When Joel Moses was working on Macsyma, the algebraic case
and the mixed algebraic-transcendental case were still open problems.

Macsyma

Comment on 2007-10-19 13:36 from Thomas Richard ( Maple Rating 2

145)

Macsyma 2.4 (the last version) returns integrate(exp(asin(x)),x) unevaluated, so even if it had the complete Risch algorithm implemented, it apparently didn't make use of it.
Anyways, that's no excuse for Maple which should return 1/2*exp(arcsin(x))*(x+sqrt(1-x^2)) (or something equivalent) without hints.

Integration

Comment on 2007-10-22 23:10 from JacquesC ( Maple Rating 4

1699)

My impression is that integration is considered an "old" topic, where Maple works "well enough", so that further work on it is not urgent or even high-priority. Medium priority or 'soonish', sure. That can translate to a few years though.

Basically, what's needed is a new(independent, third-party) review of the capabilities of multiple systems in the area of symbolic integration to be published, to rekindle this topic as something important. There were a lot of good system comparisons published in the mid 90s that fostered serious improvements. Has anyone seen good system comparisons in the last 5 years?

Integration

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