#1928 Integration of sqrt(-2*cos(t)+4*sin(t/2)^2+2)

[Anonymous]

Integration of function is not correct in (0,2*pi) - integrated function is -2^(5/2)*cos(t/2)
in: integrate(sqrt(-2*cos(t)+4*sin(t/2)^2+2),t);
out: 2^(5/2)*cos(t/2)

Maxima version: 5.20.1
Maxima build date: 21:25 12/14/2009
Host type: i686-pc-mingw32
Lisp implementation type: GNU Common Lisp (GCL)
Lisp implementation version: GCL 2.6.8

Radim

[Aleksas Domarkas]

(%i1) S:'integrate(sqrt(4-4*cos(t)),t)$

correct:
(%i2) factor(S);
(%o2) 2*integrate(sqrt(1-cos(t)),t)
(%i3) ev(%, nouns);
(%o3) 2*((-sqrt(2)*cos(t)-sqrt(2))*sin((t+%pi)/2)+sqrt(2)*sin(t)*cos((t+%pi)/2))
(%i4) trigreduce(%);
(%o4) -2^(5/2)*cos(t/2)

not correct:
(%i5) ev(S, nouns);
(%o5) 2^(5/2)*cos(t/2)

???

[Raymond Toy]

This is caused by rischint. I don't know enough about it to say more than that.

[Dan Gildea]

rischnt calls radcan1, which changes the sign of the expression.

(%o5) sqrt(4-4*cos(t))
(%i6) %,t:.1;

(%o6) .1413624380374671
(%i7) exponentialize(%o5);

(%o7) sqrt(4-2*(%e^(%i*t)+%e^-(%i*t)))
(%i8) %,t:.1;

(%o8) .1413624380374671
(%i9) radcan(%o7);

(%o9) %e^-(%i*t/2)*(sqrt(2)*%i*%e^(%i*t)-sqrt(2)*%i)
(%i10) %,t:.1;

(%o10) (.9987502603949663-.04997916927067833*%i)
*(sqrt(2)*(.09983341664682815*%i+.9950041652780258)*%i-sqrt(2)*%i)
(%i11) float(%);

(%o11) (.9987502603949663-.04997916927067833*%i)
*(1.414213562373095*(.09983341664682815*%i+.9950041652780258)*%i
-1.414213562373095*%i)
(%i12) rectform(%);

(%o12) 7.19910242530375e-17*%i-.1413624380374679

[Dan Gildea]

commit [a4e2727c317e4dd021fd035e4b1cc0bdfe405d70]

Fixes #2575.
Does not fix integrate(sqrt(4-4*cos(x)), x, -%pi, %pi) - see bug #1928.

src/defint.lisp:
o real-roots: use the roots you have, even when not all roots found