$$\sum_{k=1}^\infty \frac{\ln(4k-3)}{(4k-3)}-\frac{\ln(4k-1)}{(4k-1)}=\sum_{k}\left(\frac{\ln(4k-3)}{4k-3}-\frac{\ln(4k-1)}{4k-1} \right)[k \ge 1].$$
Let $k=j+1$. As a result,
$$4k-3=4j+4-3=4j+1, 4k-1=4j+4-1=4j+3, \text{ and } k\ge 1 \implies j \ge 0.$$
$$\sum_{k}\left(\frac{\ln(4k-3)}{4k-3}-\frac{\ln(4k-1)}{4k-1} \right)[k \ge 1]=\sum_j \left( \frac{\ln(4j+1)}{4j+1}-\frac{\ln(4j+3)}{4j+3}\right)[j \ge 0].$$
There is no 'nice' cancelling that occurs between these two summands. This is because the set $A=\{4j+1: j\ge 0\}$ and $B=\{4j+3: j \ge 0\}$ have no elements of intersection; that is, $A\cap B=\emptyset.$ While it isn't entirely relevant, this can be formally proven by noting $A$ and $B$ are equivalence classes $[1]=\{b \in \mathbb{Z}:b\equiv 1 \pmod 4\}$ and $[3]=\{b \in \mathbb{Z}: b\equiv 3\pmod 4\}$, respectively. $[1]$ and $[3]$ partition $\mathbb{Z}$ into disjoint sets, hence $A\cap B=\emptyset$. That begs the question: Where to from here?
To be honest, I don't know. I cannot conclusively prove that there is not a closed form solution for this series, but I certainly cannot find it. It does converge, and the integral test provides a nice upperbound for it.
I hope someone else can be of assistance.