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Re: An independent integration test suite
Posted:
Jul 19, 2013 3:37 PM
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clicliclic@freenet.de wrote: > > daly@axiom-developer.org schrieb: > > > > Albert, > > > > I'm getting different answers than you for some of the problems. > > In Axiom, if I do > > > > t0:=1/((1-x)^(9/2)*(1+x)^(5/2)) > > t1:=integrate(t0,x) > > t2:=D(t1,x) > > t3:=t0-t2 > > > > I get 0 as a result. However, when I take the derivative of your > > result and difference it from your initial equation I get a > > non-constant result dependent on x. > > > > Perhaps you could check your answer in other systems and let me > > know if you agree. > > > > I will be posting the first set of results shortly. > > > > This integrand appears on page 1102 of rich1b.input.pdf. Its > antiderivative > > - (8*x^5 - 16*x^4 - 4*x^3 + 24*x^2 - 9*x - 6) > /(21*(1 - x)^(7/2)*(x + 1)^(3/2)) > > is simply an algebraic function, so should be behave the same on all > systems. Indeed your expression d000 at the bottom of page 1103 > simplifies to zero for arbitrary complex x.
The result:
r0000:=1/7/((1-x)^(7/2)*(1+x)^(3/2))+1/7/((1-x)^(5/2)*(1+x)^(3/2))+_ 4/21*x/(1-x^2)^(3/2)+8/21*x/sqrt(1-x^2)
is clearly suboptimal: the integrand contains sqrt(1-x) and sqrt(x+1) so sqrt(1-x^2) should not appear in "optimal" result. > You are having more serious problems with the integrands on pages 994, > 996, 998, 1000, 1004, 1008, 1017, 1019, 1020, 1021, 1022, 1107, 1109, > and maybe elsewhere. Can FriCAS handle these (correctly) already? >
They look like variations of single example. For the first two FriCAS gets:
(5) -> integrate(1/(sqrt(2+b*x)*sqrt(6+b*x)), x)
+-------+ +-------+ log(\|b x + 2 \|b x + 6 - b x - 4) (5) - ----------------------------------- b Type: Union(Expression(Integer),...) (6) -> D(%, x)
1 (6) -------------------- +-------+ +-------+ \|b x + 2 \|b x + 6 Type: Expression(Integer) (7) -> integrate(1/(sqrt(b*x)*sqrt(4+b*x)), x)
+---+ +-------+ log(\|b x \|b x + 4 - b x - 2) (7) - ------------------------------- b Type: Union(Expression(Integer),...) (8) -> D(%, x)
1 (8) ---------------- +---+ +-------+ \|b x \|b x + 4 Type: Expression(Integer)
In general, FriCAS is supposed to handle all integrands of similar form. Sometimes for similar problems FriCAS gets very messy result (Charlwood suite has a few such examples). I am working on a cure.
-- Waldek Hebisch hebisch@math.uni.wroc.pl
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