Which integrated circuit can I use instead of multiple MOSFET?
Following Transistor's answer, you can do it in a 7400 (74HC00 or whatever).
Starting with:

First convert the OR gate into a NAND, with an inverter on each input. This actually REDUCES the gate count because you migrate the external inverter into one of the NAND inputs.
Next convert the AND into a NAND with an inverter on the output. (Which brings the gate count back to 4)
Recognising we need to feed back the AND output via an inverter into the OR gate ... just use the NAND gate and eliminate both inverters.
Giving...



simulate this circuit &ndash; Schematic created using CircuitLab