Why isn't the opamp negative input drives to 0?
The LTspice opamp is not perfectly "ideal" &mdash; instead of infinite gain, it has a very large but finite gain. This is why the feedback doesn't drive the input to exactly zero. Given that the expected output swing is &plusmn;2V/2&pi; = &plusmn;318 mV and you're measuring &plusmn;20 &micro;V, that means that the actual opamp gain is:

$$A_0 = \frac{318\text{ mV}}{20\mu\text{V}} = 15915$$

But this shouldn't make the output go "crazy" &mdash; what exactly do you think is wrong with the output?



EDIT: It bothered me that your V(n001) is a sinewave with the same phase as the input, so I dug into it a bit further. I had expected to see a voltage that was a tiny fraction of the output voltage, which should be a cosine wave with an offset.

It turns out that the default LTspice opamp.sub model is a lot less ideal than I initially thought. It not only has a finite gain of 105, but it also has a finite GBW product, which is 10 MHz by default. Internally, it is modeled as a voltage-controlled current source, with a 1&Omega; resistor across the output (a Norton source) and also a capacitor across the output that sets the GBW product according to the formula

$$C = \frac{A_{ol}}{2\pi GBW}$$

This works out to 1.59 mF using the default parameters, giving a time constant of 1.59 ms and putting the pole at 100 Hz, as expected. But it is the output impedance of 1&Omega; in series with your 1F feedback capacitor that results in the reduced overall gain and the odd phase shift.

If you substitute an actual voltage-controlled voltage source for the opamp, with a gain of 105, you get the waveform that I was expecting to see.