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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. – GEdgar Dec 3 '11 at 0:46
up vote 2 down vote accepted

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. – Srivatsan Dec 3 '11 at 0:40
The OP's integral is almost the defining integral for the trilogarithm. – J. M. Dec 3 '11 at 0:56

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