Closed form of $\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$ How would you recommend me to tackle the series 
$$\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$$?
Can we possibly express it in terms of known constants? What do you think about it?

The most compact form I was able to specify is
$$3\,\Re\left({_2F_1}\left(\begin{array}c 1,1 \\ 1+i\end{array}\middle|\,-1\right)\right) - \Im\left({_2F_1}\left(\begin{array}c 1,1 \\ 1+i\end{array}\middle|\,-1\right)\right) - 2\,\Re\left({_2F_1}\left(\begin{array}c 1,1 \\ i\end{array}\middle|\,-1\right)\right).$$
By the way for the numerical value ISC gives back the sum.
