Since $$\sum_{n=1}^{\infty } \frac{4^{n}}{n\left ( 2n+1 \right )\binom{2n}{n}} = \sum_{n=1}^{\infty } \frac{4^{n} \Gamma \left ( n \right )\Gamma \left ( n+1 \right )}{ \Gamma \left ( 2n+2 \right )} =\sum_{n=1}^{\infty } \frac{4^{n} \operatorname{B}\left ( n+1,n+1 \right )}{n},$$
then $$\sum_{n=1}^{\infty } \frac{4^{n} \operatorname{B}\left ( n+1,n+1 \right )}{n} =\int_{0}^{1}\sum_{n=1}^{\infty } \frac{4^{n}}{n} \left ( t(1-t) \right )^{n} dt.$$
Now, I've been stuck here for a day and cannot find this infinite sum.
Please give me some advice. Thank you in advance.
From the power series for $\ln (1-x)$, this is the easily evaluated$$-\int_0^1\ln (1-4t(1-t))dt=-2\int_0^1 \ln |1-2t|dt=-4\int_0^{1/2}\ln (1-2t)dt.$$