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?