Not the ordinary kind

Relevant wiki: Digamma Function

$\begin{aligned} S & = \sum_{n=1}^\infty \frac 1{n^6+n^4} & \small \color{#3D99F6} \text{By partial fraction decomposition} \\ & = \sum_{n=1}^\infty \left({\color{#3D99F6} \frac 1{n^4}} - {\color{#3D99F6}\frac 1{n^2}} + \color{#D61F06} \frac 1{n^2+1} \right) & \small \color{#3D99F6} \text{Riemann zeta function }\zeta (n) = \sum_{k=1}^\infty \frac 1{k^n} \\ & = {\color{#3D99F6} \zeta(4)} - {\color{#3D99F6}\zeta(2)} + \color{#D61F06} \sum_{k=1}^\infty \frac 1{2i} \left(\frac 1{n-i} - \frac 1{n+i} \right) & \small \color{#D61F06} \psi_0 (z+1)= - \gamma - \sum_{n=1}^\infty \left(\frac 1{z+n} - \frac 1n \right) \\ & = {\color{#3D99F6} \frac {\pi^4}{90}} - {\color{#3D99F6} \frac {\pi^2}6} + \color{#D61F06} \frac 1{2i} \left(\psi_0 (1+i) - \psi_0 (1-i)\right) & \small \color{#D61F06} \text{where }\psi_0 (\cdot) \text{ is digamma function.} \\ & = \frac {\pi^4}{90} - \frac {\pi^2}6 + \color{#D61F06} \frac 1{2i} \left(\frac 1i - \pi \cot (\pi i) \right) & \small \color{#D61F06} \psi_0 (1-z) - \psi_0(z) = \pi \cot (\pi z) \\ & = \frac {\pi^4}{90} - \frac {\pi^2}6 - \frac 12 - \frac {\pi \left(e^{-\pi} + e^\pi \right)}{2 \left(e^{-\pi} - e^\pi \right)} & \small \color{#D61F06} \psi_0 (1+z) = \psi_0(z) + \frac 1z \\ & = \frac {\pi^4}{90} - \frac {\pi^2}6 - \frac 12 + \frac {\pi \left(e^{2\pi}+1 \right)}{2 \left(e^{2\pi}-1 \right)} \\ & = \frac {\pi^4}{90} - \frac {\pi^2}6 - \frac 12 + \frac \pi 2 + \frac {\pi}{e^{2\pi}-1} \\ & = \frac 1{90} \left(\pi^4 - 15\pi^2 + 45 \pi - 45 + \frac {90\pi}{e^{2\pi}-1} \right) \end{aligned}$

Therefore, $A+B+C+D+E+F+G = 1+15+2+45+45+90+2 = \boxed{200}$ .