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Re: Integrate


"dimitris" <dimmechan at yahoo.com> wrote:
> Hello again!
>
> Of course sometimes things work quite unexpectedly!
>
> Consider again the integral (no! I am not obsessed with it!)
>
> Integrate[Log[Sin[x]^2]*Tan[x], {x, 0, Pi}]
> Integrate::idiv: Integral of Log[Sin[x]^2]*Tan[x] does not converge
> on {x,0,Pi}.

Obsessed or not, it is a curious bug, being platform dependent.

You might be interested in an antiderivative, not directly obtainable
from Mathematica AFAIK, which is valid over the whole real line:

Letting u = Abs[Cos[x]],

Integrate[Log[Sin[t]^2]*Tan[t], {t, 0, x}]

is

Log[2]^2 - Pi^2/3 + 2 Log[1 + u] Log[(1 + 1/u)/2] +
4 Log[Sqrt[2/(1 + u)]] Log[Sqrt[(1 - u)/2]] + 2 PolyLog[2, 1/(1 + u)]
+ PolyLog[2, 2 - 2/(1 + u)] + PolyLog[2, 1 - 2/(1 + u)]

I'm not sure whether the result above could be simplified further or
not.

David W. Cantrell


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