How to find this integral $I=\int_{0}^{+\infty}\frac{\{t\}(\{t\}-1)}{1+t^2}dx$?

Question:

let

$$f(t)=\int_0^t\left(\{x\}-\dfrac{1}{2}\right)dx$$ where $\{t\}$ is the fractional part of $t$,

then find this integral value

$$I=\int_0^{+\infty}\dfrac{f(t)}{1+t^2}dt$$

My try: I have $$f(t)=\int_0^t\left(\{t\}-\dfrac{1}{2}\right)dt=\dfrac{1}{2}\{t\}(\{t\}-1)$$

also can see:How find this integral $\int_{0}^{x}\left(\frac{1}{2}-\{t\}\right)dt$ so $$I=\dfrac{1}{2}\int_0^{+\infty}\dfrac{\{t\}(\{t\}-1)}{1+t^2}dt$$ so $$\sum_{n=0}^{+\infty}\int_n^{n+1}\dfrac{\{t\}(\{t\}-1)}{1+t^2}dt=\sum_{n=1}^{\infty}\int_n^{n+1}\dfrac{(t-n)(t-n-1)}{t^2+1}dt$$ since $$\int_n^{n+1}\dfrac{(t-n)(t-n-1)}{t^2+1}dt=\left(1-\dfrac{2n+1}{2}\ln{\dfrac{(n+1)^2+1}{n^2+1}}+(n^2+n-1)(\arctan{(n+1)}-\arctan({n})\right)$$ so $$I=\sum_{n=0}^{\infty}\left(1-\dfrac{2n+1}{2}\ln{\dfrac{(n+1)^2+1}{n^2+1}}+(n^2+n-1)(\arctan{(n+1)}-\arctan({n}))\right)$$

and this integral $I$ is convege, because $2f(t)=\{t\}(\{t\}-1)$ is bounded so and $$\int_0^{+\infty}\dfrac{1}{1+t^2}dt=\dfrac{\pi}{2}$$ then I can't find this sum,

Thank you very much!

The function $$f(t):={1\over2}\{t\}(\{t\}-1)$$ is periodic with period $1$ and is $\ ={1\over2}(t^2-t)$ for $0\leq t\leq1$, whence continuous. It can be developed into a Fourier series as follows: $$f(t)=-{1\over12}+\sum_{k=1}^\infty{1\over 2\pi^2 k^2}\>\cos(2k\pi t)\ .$$ Therefore we obtain $$\eqalign{I&:=\int_0^\infty{f(t)\over 1+t^2}\ dt=-{1\over12}\int_0^\infty{1\over 1+t^2}\ dt +\sum_{k=1}^\infty{1\over 2\pi^2 k^2}\int_0^\infty{\cos(2k\pi t)\over 1+t^2}\ dt\cr &\ =-{\pi\over24}+\sum_{k=1}^\infty{1\over 4\pi k^2}e^{-2\pi k}\quad .\cr}$$ Here we have used the well known integral $$\int_0^\infty{\cos(c\>t)\over 1+t^2}\ dt={\pi\over2}e^{-c}\qquad(c\geq0)\ .$$ The resulting sum cannot be expressed in elementary terms, but it is extremely well convergent. Taking the first six terms we obtain the numerical approximation $$I\doteq -0.13075101809261383373\ .$$