Math Help - Series

1. Series

Sum this series...its fairly fun

$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}n^2}{n^3+1}$

2. Hint: The answer is $\frac{1}{3}\left(1-\ln(2)+\frac{\pi}{\cosh\left(\frac{\sqrt{3}\pi}{2} \right)}\right)$. Now prove it.