/ / / pi*I\ / 5*pi*I\ / 7*pi*I\ / pi*I\ / 3*pi*I\ / 3*pi*I\\
| | | ----| -3*pi*I | ------| -pi*I | ------| pi*I | ----| 3*pi*I | ------| | ------||
| | / 7/8 8 ___\ / 7/8 8 ___ pi*I\ | 7/8 8 ___ 2 | ------- | 7/8 8 ___ 4 | ------ | 7/8 8 ___ 4 | ---- | 7/8 8 ___ 4 | ------ | 7/8 8 ___ 4 | | 7/8 8 ___ 2 ||
| | 7/8 | 3 *\/ x | 7/8 | 3 *\/ x *e | 7/8 | 3 *\/ x *e | 7/8 4 | 3 *\/ x *e | 7/8 4 | 3 *\/ x *e | 7/8 4 | 3 *\/ x *e | 7/8 4 | 3 *\/ x *e | 7/8 | 3 *\/ x *e ||
| | 63*3 *log|1 - ----------| 63*3 *log|1 - ----------------| 63*I*3 *log|1 - ----------------| 63*3 *e *log|1 - ------------------| 63*3 *e *log|1 - ------------------| 63*3 *e *log|1 - ----------------| 63*3 *e *log|1 - ------------------| 63*I*3 *log|1 - ------------------||
| 2 | 7 \ 3 / \ 3 / \ 3 / \ 3 / \ 3 / \ 3 / \ 3 / \ 3 /|
|2*x *|- ---------- - --------------------------- + --------------------------------- - ----------------------------------- - -------------------------------------------- - ------------------------------------------- - --------------------------------------- - ------------------------------------------- + -------------------------------------|
| | 16*x 7/8 7/8 7/8 7/8 7/8 7/8 7/8 7/8 |
| | -16 + ---- 128*x 128*x 128*x 128*x 128*x 128*x 128*x 128*x |
| \ 3 / |x|
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- for --- < 1
< 63 3
|
| oo
| ____
| \ `
| \ -n n
| \ n*3 *x
| / -------- otherwise
| / -9 + 8*n
| /___,
| n = 2
\
$$\begin{cases} \frac{2 x^{2} \left(- \frac{7}{\frac{16 x}{3} - 16} - \frac{63 \cdot 3^{\frac{7}{8}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} e^{\frac{i \pi}{4}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{i \pi}{4}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} i \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{i \pi}{2}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} e^{\frac{3 i \pi}{4}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{3 i \pi}{4}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} + \frac{63 \cdot 3^{\frac{7}{8}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{i \pi}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} e^{- \frac{3 i \pi}{4}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{5 i \pi}{4}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} + \frac{63 \cdot 3^{\frac{7}{8}} i \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{3 i \pi}{2}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}} - \frac{63 \cdot 3^{\frac{7}{8}} e^{- \frac{i \pi}{4}} \log{\left(- \frac{3^{\frac{7}{8}} \sqrt[8]{x} e^{\frac{7 i \pi}{4}}}{3} + 1 \right)}}{128 x^{\frac{7}{8}}}\right)}{63} & \text{for}\: \frac{\left|{x}\right|}{3} < 1 \\\sum_{n=2}^{\infty} \frac{3^{- n} n x^{n}}{8 n - 9} & \text{otherwise} \end{cases}$$
Piecewise((2*x^2*(-7/(-16 + 16*x/3) - 63*3^(7/8)*log(1 - 3^(7/8)*x^(1/8)/3)/(128*x^(7/8)) + 63*3^(7/8)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(pi*i)/3)/(128*x^(7/8)) - 63*i*3^(7/8)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(pi*i/2)/3)/(128*x^(7/8)) - 63*3^(7/8)*exp(-3*pi*i/4)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(5*pi*i/4)/3)/(128*x^(7/8)) - 63*3^(7/8)*exp(-pi*i/4)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(7*pi*i/4)/3)/(128*x^(7/8)) - 63*3^(7/8)*exp(pi*i/4)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(pi*i/4)/3)/(128*x^(7/8)) - 63*3^(7/8)*exp(3*pi*i/4)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(3*pi*i/4)/3)/(128*x^(7/8)) + 63*i*3^(7/8)*log(1 - 3^(7/8)*x^(1/8)*exp_polar(3*pi*i/2)/3)/(128*x^(7/8)))/63, |x|/3 < 1), (Sum(n*3^(-n)*x^n/(-9 + 8*n), (n, 2, oo)), True))