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I tried entering "inverse fourier transform of (I-x)^3*Log[I-x]^4" at http://www.wolframalpha.com/ and it timed out. I think I read somewhere that Wolfram have ensured that Alpha knows all formulae from Gradshteyn and Ryzhik, as well as many other things. Similarly, Maple just gives me an answer in terms of the derivatives of the inverse Fourier transform of $\log(i-x)^4$, so I suspect that there is no explicit answer. I would certainly say that you are more likely to get a useful answer out of Maple or Mathematica than out of a book of tables.