I am just guessing but in a single supply system, there are no negative voltages. So, why does your square wave not center around 2.5V and swing from 0V to 5V.
Many lo-end audio circuits are exactly like that - single supply, input capacitor, signal centered around ground ( 0V ).
True, but they usually bias the input pin at Vcc/2. The problem with the circuit above isn't so much a matter of bias as the fact that the signal can go negative, below ground, and the integrator can't follow along.
I don't see how adding bias to the input pin will work with the signal going below ground and I don't see how the bias will work at all without affecting the input - the reason the capacitor was put there in the first place, I suppose.
I would like to see what happens when the input signal is changed to 0-5V and the input capacitor removed. But not enough to lay out the circuit from scratch.
Analog computing, my main interest in op amps, is usually done with integrators using dual supply op amps but I don't see why it couldn't be done with single supply op amps just as well. It's just a case of Vcc/2 is the new 0V.
I also don't see how the single supply design can get to the rails because rail-to-rail is a marketing term, not an engineering fact. In an analog computer scenario, we would probably leave a volt or so off of each rail so, for a Vcc of 5V, the signals would vary between 1V..4V and the center would still be at 2.5V
ETA: I would adjust the signal source to center on 2.5 and have a 3V peak to peak amplitude.
For the old 741 integrators, the supply was +-15V and the signals were constrained to +-10V.