[Maxima-bugs] [ maxima-Bugs-3405408 ] integrate bug

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Bugs item #3405408, was opened at 2011-09-07 12:31
Message generated for change (Comment added) made by aleksasd
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Category: None
Group: None
Status: Open
Resolution: None
Priority: 5
Private: No
Submitted By: Aleksas (aleksasd)
Assigned to: Nobody/Anonymous (nobody)
Summary: integrate bug

Initial Comment:
 Wrong:
(%i1) integrate(acos((x-1/2)/sqrt(1-x^2)),x);
(%o1) -(%pi*x)/2

 The result should be
(%i2) sol:(2^(3/2)*x*acos((2*x-1)/(2*sqrt(1-x^2)))
-sqrt(2)*asin((10*x+1)/(2*sqrt(7)*(x+1)))
+sqrt(2)*asin((6*x-5)/(2*sqrt(7)*(x-1)))
+asin((4*x-1)/sqrt(7)))/2^(3/2)$
 Test of sol:
(%i3) f:acos((x-1/2)/sqrt(1-x^2))$
(%i4) diff(sol,x)-f,radcan;
(%o4) 0
(%i5) is(%=0);
(%o5) true

 Next a detailed solution has bug:
(%i6) load(bypart)$
(%i7) assume(abs(x)<1);
(%o7) [abs(x)<1]
(%i8) byparts(f,x,f,x),factor$
(%i9) sol1:ev(%, nouns),factor$
 Test of sol1:
(%i10) diff(sol1,x)-f,radcan$
(%i11) is(%=0);
(%o11) false

 Note: this integral is from 
http://www.math.utexas.edu/pipermail/maxima/2011/025866.html
(%i12) S1:'integrate(acos((2*c+k*u)/sqrt((1-k^2/4)*(1-u^2))),u)$
(%i13) subst([k=2/sqrt(5),c=-1/2/sqrt(5)],S1),factor;
(%o13) integrate(acos((2*u-1)/(2*sqrt(1-u^2))),u)

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Comment By: Aleksas (aleksasd)
Date: 2011-09-10 08:44

Message:
Correct value of
integrate(acos((x-1/2)/sqrt(1-x^2)),x)
and "radcan" bug

(%i1) f:acos((x-1/2)/sqrt(1-x^2))$
(%i2) load(bypart)$
(%i3) assume(abs(x)<1)$
(%i4) byparts(f,x,f,1),factor$
(%i5) sol2:ev(%, nouns),factor;
(%o5) (2^(3/2)*x*acos((2*x-1)/(2*sqrt(1-x^2)))
-sqrt(2)*asin((10*x+1)/(2*sqrt(7)*(x+1)))
-sqrt(2)*asin((6*x-5)/(2*sqrt(7)*(x-1)))
+asin((4*x-1)/sqrt(7)))/2^(3/2)

 Test of solution(the first method):
(%i6) diff(sol2,x)-f$ factor(%);
(%o7) 0
(%i8) is(%=0);
(%o8) true

 Test of solution(the second method):
(%i9) diff(sol2,x)-f$ radcan(%);
(%o10) 1/((%i*x-%i)*sqrt(8*x^2-4*x-3))
(%i11) is(%=0);
(%o11) false

 I think that solution sol2 is correct
and "radcan" has bug

(%i12) build_info()$
Maxima version: 5.25.0
Maxima build date: 12:0 8/2/2011
Host type: i686-pc-mingw32
Lisp implementation type: Clozure Common Lisp
Lisp implementation version: Version 1.7-r14925M  (WindowsX8632)

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Comment By: Aleksas (aleksasd)
Date: 2011-09-07 13:11

Message:
WolframAlpha solution of this problem is not correct.
The result is undefined for x=0 (in real domain).

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