Wolfram Alpha claims there is no closed form in terms of standard funcions for $\int \lfloor x \rfloor dx$ but we believe we found simple closed form agreeing with experimental data.
Define $i_1(x)=x - \frac12 i \frac{\log{(-e^{-2 i \pi x} )}}{\pi} - \frac12$
For real $x$ we have $\lfloor x \rfloor=i_1(x)$ where $\log$ means the principal branch of the logarithm.
We have $\int \lfloor x \rfloor dx= \int i_1(x) dx=I_1(x)=\frac{1}{2} \, x^{2} - \frac{1}{2} \, x + \frac{\log\left(-e^{\left(-2 i \, \pi x\right)}\right)^{2}}{8 \, \pi^{2}}$
To get rid of complex number, one may use $i_2(x)=x + \frac{\arctan\left(\cot\left(\pi x\right)\right)}{\pi} - \frac{1}{2}$, but sagemath dislikes division by zero, which works fine in mpmath.
Q1 Is this closed form correct, do branches of $\log$ cause problem?
Q2 Why Wolfram Alpha claims there is no closed form?
Added: In answer Carlo Beenakker gave simpler closed form. It is consistent with our result up to a constant of integration.
