Is there a closed form for $$\underbrace{\frac{1}{a+1}+\frac{2a}{a^2+1}+\frac{3a^3}{a^4+1}+\frac{4a^7}{a^8+1}+\cdots}_{n \text{ terms}} $$ where $|a|<1$? Written in sigma notation, the series is $$\sum _{r=0}^{n-1} \frac{(r+1) a^{2^r-1}}{a^{2^r}+1}$$
Could someone help me how to go further? Thanks.