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Is there a series representation for $\frac{1}{\log(x)}$?

for $x>0$

it is known that:
$$\log(x)=2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}$$
Is there a series representation for $\frac{1}{\log(x)}$ in the following form?
$$\frac{1}{\log(x)}=a_1(x)+a_2(x)+a_3(x)+\dots=\sum_{k=1}^\infty a_k(x)$$
Or, is there a way to transform the expression
$$ \frac{1}{2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}}$$
into the aforementioned form?

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