Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result :
$$\sum\_{n=0}^{\infty}\frac{1}{n^2+1}= \frac{1+ \pi\coth\pi}{2}$$
is known by WolframAlpha. But my question is : does there exist a general theory of similar results (something like exact formulas for $\sum\_{n=0}^{+\infty}\frac{1}{an^2+bn+c}$, at least when $a$, $b$ and $c$ are integers)?



Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$
To do $\sum\frac{1}{an^2+bn+c}$, factor the denominator and get a [digamma](https://en.wikipedia.org/wiki/Digamma_function) answer.
$$
\sum\_{n=0}^\infty \frac{1}{(x+p)(x+q)} = \frac{\psi(p)-\psi(q)}{p-q}
$$
And note that digamma of a rational can be evaluated in terms of logarithms and trig functions ([Gauss's digamma theorem](https://en.wikipedia.org/wiki/Digamma_function#Gauss.27s_digamma_theorem)).
$$
\sum\_{n=0}^\infty\frac{1}{(n+\frac{1}{4})(n+\frac{1}{3})} =
36\log 2+6\pi-2\pi\sqrt{3}-18\log 3 .
$$


