nob...@nowhere.invalid
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Sam Blake schrieb:
>
> For fun I have been implementing the rational integral routines from
> Bronstein's Symbolic Integration I in Mathematica. As part of my
> testing I stumbled across the following example integral
>
> \int x/(2 + 4 x + 5 x^2 + 2 x^3 + x^4) dx
>
> Rubi does well on this example
>
> In[66]:= Int[x/(2 + 4 x + 5 x^2 + 2 x^3 + x^4), x]
>
> Out[66]= -(ArcTan[(1 + 2 x)/Sqrt[7 - 4 Sqrt[2]]]/(
> 2 Sqrt[2 (7 - 4 Sqrt[2])])) +
> ArcTan[(1 + 2 x)/Sqrt[7 + 4 Sqrt[2]]]/(2 Sqrt[2 (7 + 4 Sqrt[2])]) -
> ArcTanh[(7 + 4 (1/2 + x)^2)/(4 Sqrt[2])]/(2 Sqrt[2])
>
> While my rational function integrator returns
>
> In[67]:= IntegrateRational[x/(2 + 4 x + 5 x^2 + 2 x^3 + x^4), x]
>
> Out[67]= 1/2 (((-1 + I Sqrt[7 - 4 Sqrt[2]]) Log[
> 1/2 (1 - I Sqrt[7 - 4 Sqrt[2]]) + x])/(
> 2 (2 + 5/2 (-1 + I Sqrt[7 - 4 Sqrt[2]]) +
> 3/4 (-1 + I Sqrt[7 - 4 Sqrt[2]])^2 +
> 1/4 (-1 + I Sqrt[7 - 4 Sqrt[2]])^3)) + ((-1 -
> I Sqrt[7 - 4 Sqrt[2]]) Log[
> 1/2 (1 + I Sqrt[7 - 4 Sqrt[2]]) + x])/(
> 2 (2 + 5/2 (-1 - I Sqrt[7 - 4 Sqrt[2]]) +
> 3/4 (-1 - I Sqrt[7 - 4 Sqrt[2]])^2 +
> 1/4 (-1 - I Sqrt[7 - 4 Sqrt[2]])^3)) + ((-1 +
> I Sqrt[7 + 4 Sqrt[2]]) Log[
> 1/2 (1 - I Sqrt[7 + 4 Sqrt[2]]) + x])/(
> 2 (2 + 5/2 (-1 + I Sqrt[7 + 4 Sqrt[2]]) +
> 3/4 (-1 + I Sqrt[7 + 4 Sqrt[2]])^2 +
> 1/4 (-1 + I Sqrt[7 + 4 Sqrt[2]])^3)) + ((-1 -
> I Sqrt[7 + 4 Sqrt[2]]) Log[
> 1/2 (1 + I Sqrt[7 + 4 Sqrt[2]]) + x])/(
> 2 (2 + 5/2 (-1 - I Sqrt[7 + 4 Sqrt[2]]) +
> 3/4 (-1 - I Sqrt[7 + 4 Sqrt[2]])^2 +
> 1/4 (-1 - I Sqrt[7 + 4 Sqrt[2]])^3)))
>
> I find similar results from AXIOM and FriCAS. Is this a limitation of
> Rioboo's algorithm?
>
This is an interesting example. Derive 6.10 also solves the integral in
real terms:
INT(x/(2 + 4*x + 5*x^2 + 2*x^3 + x^4), x)
SQRT(238 - 136*SQRT(2))*ATAN(SQRT(119 - 68*SQRT(2))*(2*x + 1)/17)/68
- SQRT(136*SQRT(2) + 238)*ATAN(SQRT(68*SQRT(2) + 119)*(2*x + 1)/17)/68
+ SQRT(2)*LN((x^2 + x - SQRT(2) + 2)/(x^2 + x + SQRT(2) + 2))/8
as it starts by factoring the denominator:
2 + 4*x + 5*x^2 + 2*x^3 + x^4 =
(x^2 + x + SQRT(2) + 2)*(x^2 + x - SQRT(2) + 2)
and then expands the integrand into partial fractions. Indeed, the
cubic resolvent of the denominator factors as (y - 4)*(y^2 - y - 4).
By contrast, FriCAS 1.3.9 returns a whopping:
((34^(1/2)*((-102)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)^2+(-68)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+((-102)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+5))^(1/2)+((-34)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+(-34)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)))*log(((164*34^(1/2)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+38*34^(1/2))*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+(38*34^(1/2)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+5*34^(1/2)))*((-102)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)^2+(-68)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+((-102)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+5))^(1/2)+((5576*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+1292)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)^2+(5576*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(-170))*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+(1292*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(-170)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(52*x+(-42)))))+(((-1)*34^(1/2)*((-102)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)^2+(-68)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+((-102)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+5))^(1/2)+((-34)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+(-34)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)))*log((((-164)*34^(1/2)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(-38)*34^(1/2))*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+((-38)*34^(1/2)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(-5)*34^(1/2)))*((-102)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)^2+(-68)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+((-102)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+5))^(1/2)+((5576*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+1292)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)^2+(5576*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(-170))*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+(1292*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(-170)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(52*x+(-42)))))+(68*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)*log(((-11152)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(-2584))*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)^2+((-11152)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+340)*rootOf((136*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+136*%%E1*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(136*%%E1^2+(-5))*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(136*%%E1^3+(-5)*%%E1+1))/136,%%E1)+((-11152)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+410*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(52*x+53)))+68*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)*log(11152*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^3+(-2584)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)^2+(-70)*rootOf((544*%%E0^4+(-20)*%%E0^2+4*%%E0+1)/544,%%E0)+(52*x+135)))))/68
which is the sum of three logarithms involving many nested rootOf()s
where the %%En denote local variables. The quartic of the inner
rootOf() factors as:
544*z^4 - 20*z^2 + 4*z + 1 =
1/17*(68*SQRT(2)*z^2 + 34*z + 3*SQRT(2) - 1)
*(68*SQRT(2)*z^2 - 34*z + 3*SQRT(2) + 1)
and its cubic resolvent is (34*y - 3)*(272*y^2 + 34*y + 1).
Let's see if FriCAS version 1.3.10 will do better.
Martin.