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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.2019-03-21

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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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    The OP's integral is almost the defining integral for the trilogarithm.2011-12-03