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Re: Goursat pseudo-elliptics and the Wolfram Integrator
Posted:
Dec 17, 2012 12:41 PM
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clicliclic@freenet.de wrote:
> Example 2 (quartic radicand):
>
> Integrate[(k*x^2 - 1)/((a*k*x + b)*(b*x + a)
> *Sqrt[(1 - x^2)*(1 - k^2*x^2)]), x]
>
> ... "Mathematica could not find a formula for your integral. Most likely
> this means that no formula exists." Waouw! Here the elementary
> antiderivative is:
>
> 2/(Sqrt[(a + b)*(a*k + b)]*Sqrt[(a - b)*(a*k - b)])
> *ArcTanh[Sqrt[(a + b)*(a*k + b)]*Sqrt[(1 - x^2)*(1 - k^2*x^2)]
> /(Sqrt[(a - b)*(a*k - b)]*(1 - x)*(1 - k*x))]
>
> The theory behind these integrals is given in: Edouard Goursat, Note sur
> quelques int?grales pseudo-elliptiques, Bulletin de la Soci?t?
> Math?matique de France 15 (1887), 106-120, on-line at:
>
> <http://www.numdam.org/item?id=BSMF_1887__15__106_1>
>
> This was written 125 years ago - apparently too recent for the "Risch"
> integrator of Mathematica 8. I expect that FriCAS can do the second
> integral too.
FriCAS result:
integrate((k*x^2 - 1)/((a*k*x + b)*(b*x + a)*sqrt((1 - x^2)*(1 - k^2*x^2))), x)
(1)
[
log
4 2 6 4 2 4 4 2 3 2 4 4 2 2
(2a b - 2a )k + (2a b - 2a b )k + (- 2a b + 2a b )k
+
6 2 4
(- 2b + 2a b )k
*
2
x
+
3 3 5 4 3 3 5 3
(2a b - 2a b)k + (4a b - 4a b)k
+
5 3 3 5 2 5 3 3 5
(- 2a b + 4a b - 2a b)k + (- 4a b + 4a b )k - 2a b
+
3 3
2a b
*
x
+
4 2 6 3 2 4 4 2 2 2 4 4 2 6
(2a b - 2a )k + (2a b - 2a b )k + (- 2a b + 2a b )k - 2b
+
2 4
2a b
*
+-----------------------+
| 2 4 2 2
\|k x + (- k - 1)x + 1
+
2 2 4 4 2 2 3 4 2 2 2 4
((a b - 2a )k - 2a b k + (- 2b + a b )k )x
+
3 4 3 3 3 3 3 2
- 2a b k + (- 2a b - 4a b)k + (- 4a b - 2a b)k
+
3
- 2a b k
*
3
x
+
2 2 4 4 2 2 4 3
(- 2a b + a )k + (- 4a b - 2a )k
+
4 2 2 4 2 4 2 2 4 2 2
(b - 12a b + a )k + (- 2b - 4a b )k + b - 2a b
*
2
x
+
3 3 3 3 2 3 3 3
(- 2a b k + (- 2a b - 4a b)k + (- 4a b - 2a b)k - 2a b )x
+
2 2 4 2 2 2 4 2 2
(a b - 2a )k - 2a b k - 2b + a b
*
+---------------------------+
| 2 2 4 2 4 2 2
\|(- a b + a )k + b - a b
/
2 2 2 4 3 2 3 3 4 2 2 2 4 2
a b k x + (2a b k + 2a b k)x + (a k + 4a b k + b )x
+
3 3 2 2
(2a b k + 2a b )x + a b
/
+---------------------------+
| 2 2 4 2 4 2 2
2\|(- a b + a )k + b - a b
,
+-------------------------+ +-----------------------+
| 2 2 4 2 4 2 2 | 2 4 2 2
\|(a b - a )k - b + a b \|k x + (- k - 1)x + 1
atan(------------------------------------------------------)
2 2 2 2 2 2 2
(a k + b k)x + (a b k + 2a b k + a b)x + a k + b
------------------------------------------------------------]
+-------------------------+
| 2 2 4 2 4 2 2
\|(a b - a )k - b + a b
There are two alternatives, one in terms of 'log', the other
(shorter) in terms of 'atan'.
--
Waldek Hebisch
hebisch@math.uni.wroc.pl
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