Primitive of Reciprocal of Logarithm of x

Theorem

 * $\displaystyle \int \frac {\d x} {\ln x} = \map \ln {\ln x} + \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {\ln x}^k} {k \times k!} + C$

Proof
From Primitive of $\dfrac {x^m} {\ln x}$:
 * $\displaystyle \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$

The result follows by setting $m = 1$.

Also see

 * Primitive of $\dfrac 1 {\ln x}$ has no Solution in Elementary Functions