Elliptic integrals (Was Re: Missing resimplify in lambert_w)

Michel Talon wrote:
> Billinghurst, David (RTATECH) wrote:
>
>   
>>> From: Dieter Kaiser
>>>       
>> Great.  I am happy to do some detailed coding and testing for
>> special functions.  I was looking at your new code last night
>> - trying to work out how to integrate Jacobi elliptic functions.
>> I didn't find a neat, general solution but it looked feasible.
>>
>> I was also wondering how to add a hook/property to integrate
>> (simple) expressions containing a special function, such as:
>>  * integrate( sin(x)/x, x)
>>  * integrate( x^(n+1).bessel_j(n,x), x)
>>  * integrate( jacobi_sn(u,m)^n, dx)
>>
>>     
>
> Well, speaking of elliptic functions, i think a worthy addition would be
> recognize the fact that any integral of the form 
> integrate(R(y(x),x),x)
> where R(y,x) is a rational function of x and y, and y(x) is the square root
> of a polynomial in x of degree 3 or 4 without double roots is  elliptic.
>   
I have written some maxima code to do some elliptic integrals.  It could
do elliptic integrals of the first and second type, but not always.  And
I could never decide exactly which Jacobian elliptic function to use, so
the routines would return both functions.   And it needed some help from
the user to recognize the integrand; it didn't do a very good job of
recognizing the integrand.  For example sqrt((1-x^2)/(1-k^2*x^2)) could
be written as (1-x^2)/sqrt((1-x^2)*(1-k^2*x^2)), which is desired form.

I keep meaning to finish it but never have.  Any one have it if interested.


Ray