On Thu, Dec 21, 2023 at 08:32:46PM +0800, Qian Yun wrote:
> Off topic a little bit:
>
> For parametric integrals, do you think these are good ideas?
>
> 1. Integration by substitution.
>
> For 1/(a^5+x^5), recognize and transform it to 1/a^5*1/(1+(x/a)^5).
> Substitute x/a with y.
If you literally mean cases like 1/(a^n + x^n), then this is probably
too special. If you want more general cases, than is becomes
more tricky. For example a lot of integrals in Rubi testsute
contains 'a*x + b' as a factors. One could try substitution
'y = a*x + b', but this may complicate other parts of the integral.
Currently FriCAS useses substitution when they give "substantial"
simplification:
- allow completely getting rid of a transcendental kernel
- reduce degree of algebriac kernels
> 2. partialFraction instead of current resultant method.
>
> e.g. integrate(1/((x-a)*(x-b)*(x-c)*(x-d)),x)
That is promising idea. In principle partialFraction should
always be faster than resultant and AFAICS we will not
loose elementary integrability by splitting into partial fractions.
But it needs testing.
--
Waldek Hebisch
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