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/CAKgW%3D6L6z7-wxMu9nTJEeoTdgvJSp93hOtjzct80k33Zu96%3DyQ%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
