HINT :
$$ f(x)=\sum_{n=1}^\infty \frac{(-1)^n}{(n\pi)^3}\sin(n\pi x)$$
$$ f''(x)=-\sum_{n=1}^\infty \frac{(-1)^n}{n\pi}\sin(n\pi x)$$
$\sin(n\pi (x-1))=\sin(n\pi x)\cos(n\pi)=(-1)^n\sin(n\pi x)$
$$ f''(x)=-\sum_{n=1}^\infty \frac{1}{n\pi}\sin(n\pi (x-1))$$
With $X=x-1$ we get the Sawtooth wave or Sawtooth function $SW(X)$. http://mathworld.wolfram.com/FourierSeriesSawtoothWave.html
$$\frac12-\sum_{n=1}^\infty \frac{1}{n\pi}\sin(n\pi X)=SW(X).$$
$$f''(x)=-\frac12+SW(\pi (x-1))$$
I suppose that you can continue the double integration on the first segment which gives the repeated pattern :
This looks like a sinusoid, but it isn't a sinusoid. The pattern is an arc of cubic curve which equation is obtained thanks to the double integration of the linear function.
Note : The Sawtooth function can be expressed on the form of a complex equation. See Eq.$(11)$ in the web-page referenced above. Theoretically the integrations allows to express $f(x)$ on closed form in terms of complex polylogarithm functions.