# [Maxima] Is integration and differentiation code sepparate?

Karl-Dieter Crisman kcrisman at gmail.com
Fri Nov 12 09:56:00 CST 2010

```> I've been trying to automatically develop a large number of tests for Sage.
> Hopefully millions of them.
>
> Some of those tests actually test Maxima. I've only just started this, but the
> approach I've taken to date is:
>
> 1) Generate a "random" polynomial using a pseudo random number generator.
> 2) Simplify the polynomail, so the output is in a form that's reproducible.
> 3) Integrate the polynomial.
> 4) Differentiate the result from (3)
> 5) Simplify the result from (4)
> 6) Compare the results from (2) and (5).
>
> All being well, steps (2) and (5) should be equal.
>
> For simple polynomails, this does indeed seem to be the case. I've generated
> many thousands of them, sometimes with complex coefficients and complex powers,
> and all seems to be ok.
>
> I'm just wondering if there's any common code, which might actually mean the
> such tests are meaningless.
>
>
>
> I've found more complex cases (using sin, sinh, tan arctan etc) where the
> results of integrating a function, then differentiating it do NOT lead to back
> to the original function using Maxima.
>
> But I've tried the same equations in Mathematica too, and that also has
> problems, with the result appearing to be much more complicated than what I
> started with. Perhaps I'm just expecting too much. Perhaps this approach is not
> mathematically valid - I'm not a mathematician.
>
>
> Dave
>
>
>

Just to say something (fairly) obvious to mathematicians, sometimes
there are antiderivatives that are equal up to a constant (or even
equal) which don't look anything alike - and which presumably it's
very hard to automatically check if they are equal up to a constant.
This happens particularly often with things involving combinations of
trig functions, but also with things like arcsinh/ln(stuff) and other
integrals of radicals... I bet looking at a good table of integrals
would show some of these things.    So that even calc textbooks point
this out, that using different systems will give you different (but