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sum of integrals over patial intervals != integral over whole interval


Dear group,

I wanted Mathematica to show, that for f[x_]:=Log[Sin[x]^2]Tan[x], 
Integrate[f[x],{x,0,Pi}]==0, because f[x]+f[Pi-x]==0.

Mathematica says Integrate[f[x],{x,0,Pi}] does not converge, but 
Integrate[f[x],{x,0,Pi/2}] and Integrate[f[x],{x,Pi/2,Pi}] evaluate to 
-Pi^2/12 resp. P^2/12 and the sum is zero. The more general integral 
Integrate[f[x],{x,0,z},Assumptions->Pi/2<z<=Pi] evaluates explicitly (?).

What did I do wrong?
http://people.freenet.de/Peter_Berlin/Mathe/komisch.nb

TIA,
Peter


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