https://github.com/sympy/sympy/issues/9723
On Thursday, July 23, 2015 at 3:07:36 PM UTC-5, Kalevi Suominen wrote: > > Surprising, but satisfactory. I had forgotten that the meijerg method > would work for some elliptic integrals, though in this case none of those > remain in the solution. > > On Thursday, July 23, 2015 at 10:50:43 PM UTC+3, Aaron Meurer wrote: >> >> Oh, I take that back: >> >> In [16]: print(integrate(sqrt(sqrt(x))*sqrt(1 + sqrt(x)))) >> 2*x**(7/4)/(3*sqrt(sqrt(x) + 1)) + 5*x**(5/4)/(6*sqrt(sqrt(x) + 1)) - >> x**(3/4)/(12*sqrt(sqrt(x) + 1)) - x**(1/4)/(4*sqrt(sqrt(x) + 1)) + >> asinh(x**(1/4))/4 >> >> (this is coming from the meijerg algorithm). And I wouldn't be surprised >> if some rewriting of it made heurisch handle it as well. The real power of >> the full Risch algorithm is that it uses canonical forms of algebraic >> functions, so it is immune to simple rewrites like this one. >> >> Aaron Meurer >> >> >> On Thu, Jul 23, 2015 at 2:45 PM, Aaron Meurer <[email protected]> wrote: >> >>> It's worth pointing out that computing this integral is very likely >>> going to require the algebraic Risch algorithm. Maybe a pattern matching >>> case for it can be added to something like Rubi or manulaintegrate (I'm >>> actually curious if Rubi can do this), but the solution, at least as Sage >>> has presented it, looks rather complicated (e.g., the fact that the result >>> has logarithms means that pattern matching integrators are going to be >>> limited without at least *some* Risch algorithm). >>> >>> Aaron Meurer >>> >>> On Thu, Jul 23, 2015 at 2:42 PM, Aaron Meurer <[email protected]> wrote: >>> >>>> I think it is correct, at least algebraically: >>>> >>>> In [6]: a = S("-1/12*(3*(sqrt(x) + 1)^(5/2)/x^(5/4) - 8*(sqrt(x) + >>>> 1)^(3/2)/x^(3/4) - 3*sqrt(sqrt(x) + 1)/x^(1/4))/((sqrt(x) + 1)^3/x^(3/2) - >>>> 3*(sqrt(x) + 1)^2/x + 3*(sqrt(x) + 1)/sqrt(x) - 1) + 1/8*log(sqrt(sqrt(x) >>>> + >>>> 1)/x^(1/4) + 1) - 1/8*log(sqrt(sqrt(x) + 1)/x^(1/4) - 1)") >>>> >>>> In [7]: print(simplify(a.diff(x))) >>>> x**(1/4)*sqrt(sqrt(x) + 1) >>>> >>>> (if you distribute sqrt(sqrt(x)) inside the sqrt(sqrt(x) + 1) you get >>>> sqrt(x + sqrt(x)) >>>> >>>> Aaron Meurer >>>> >>>> On Thu, Jul 23, 2015 at 2:37 PM, Kalevi Suominen <[email protected]> >>>> wrote: >>>> >>>>> >>>>> >>>>> On Thursday, July 23, 2015 at 8:31:41 PM UTC+3, Denis Akhiyarov wrote: >>>>>> >>>>>> Sympy cannot do this? >>>>>> >>>>>> integrate(sqrt(x+sqrt(x))) >>>>>> >>>>>> BTW, this is computed by SAGE: >>>>>> >>>>>> -1/12*(3*(sqrt(x) + 1)^(5/2)/x^(5/4) - 8*(sqrt(x) + 1)^(3/2)/x^(3/4) >>>>>> - 3*sqrt(sqrt(x) + 1)/x^(1/4))/((sqrt(x) + 1)^3/x^(3/2) - 3*(sqrt(x) + >>>>>> 1)^2/x + 3*(sqrt(x) + 1)/sqrt(x) - 1) + 1/8*log(sqrt(sqrt(x) + >>>>>> 1)/x^(1/4) + >>>>>> 1) - 1/8*log(sqrt(sqrt(x) + 1)/x^(1/4) - 1) >>>>>> >>>>> >>>>> >>>>> I Is it possible to verify that this really is a solution. The >>>>> integrand seems to belong to an elliptic function field where integration >>>>> rarely succeeds in terms of algebraic expressions. >>>>> >>>>> >>>>> -- >>>>> You received this message because you are subscribed to the Google >>>>> Groups "sympy" group. >>>>> To unsubscribe from this group and stop receiving emails from it, send >>>>> an email to [email protected]. >>>>> To post to this group, send email to [email protected]. >>>>> Visit this group at http://groups.google.com/group/sympy. >>>>> To view this discussion on the web visit >>>>> https://groups.google.com/d/msgid/sympy/f09ea601-6539-4d3e-9eb9-636d238882df%40googlegroups.com >>>>> >>>>> <https://groups.google.com/d/msgid/sympy/f09ea601-6539-4d3e-9eb9-636d238882df%40googlegroups.com?utm_medium=email&utm_source=footer> >>>>> . >>>>> >>>>> For more options, visit https://groups.google.com/d/optout. >>>>> >>>> >>>> >>> >> -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/78bfda11-0575-4668-910e-487940de50d3%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
