An integral which returns an expression in terms
of the dilogarithm function Li[2](z), in which z is
in general complex, is
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(%i11) integrate(log(sin(x)),x,1,2);
(%o11) -2*%i*atan(sin(2)/(cos(2)+1))-2*%i*atan(sin(2)/(cos(2)-1))
+2*log(sin(2))-log(2*cos(2)+2)
-log(2-2*cos(2))
+%i*atan(sin(1)/(cos(1)+1))
+%i*atan(sin(1)/(cos(1)-1))-log(sin(1))
+log(2*cos(1)+2)/2+log(2-2*cos(1))/2
+%i*li[2](%e^(2*%i))+%i*li[2](-%e^(2*%i))
-%i*li[2](%e^%i)-%i*li[2](-%e^%i)+3*%i/2
---------------------------------------------
quadpack gives a numerical value -0.0455 for this integral,
but maxima doesn't know how to get numbers for Li[2](z)
if z is complex
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(%i12) float(li[2](1+%i));
(%o12) li[2](%i+1.0)
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wolframalpha gives (0.62 + 1.5 i) approx
for this float (using PolyLog[2, 1 + I]).
Ted Woollett