There is no real need for an especially good OP in this circuit. The JFET should already give enough gain, so that something like a NE5532 or OP27 should be enough. It's also possible that the LEDs give more noise than the OP.
The high impedance input stage gives high noise levels near it's cut of. So there likely needs to follow a second high pass filter that sets the actually used lower frequency limit. With this second filter likely coming anyway, there is no need for the second version with extra integration. Also there is no need to make the inverter stage especially high impedance and thus high noise. So it would be better to use 2 more conventional 10 K Resistors.