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Is there a primitive function to:

$$\int \! \frac{\int \! \frac{\ln(x+1)\, \mathrm{d} x}{x}\, \mathrm{d} x}{x}$$

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    Yes, any continuous function has a primitive (=antiderivative) by the fundamental theorem of calculus.2011-12-03

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Is a polylogarithm a primitive function?

This looks something like $-\operatorname{Li}_3(-x)+k_2+k_1 \ln x$.

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    I get a feeling that the OP used the term "primitive" function in the sense of an antiderivative, rather than an "elementary" function.2011-12-03
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    The OP's integral is almost the defining integral for the trilogarithm.2011-12-03