By applying entropy codecs with learned data distributions, neural compressors have significantly outperformed traditional codecs in terms of compression ratio. However, the high inference latency of neural networks hinders the deployment of neural compressors in practical applications. In this work, we propose Integer-only Discrete Flows (IODF) an efficient neural compressor with integer-only arithmetic. Our work is built upon integer discrete flows, which consists of invertible transformations between discrete random variables. We propose efficient invertible transformations with integer-only arithmetic based on 8-bit quantization. Our invertible transformation is equipped with learnable binary gates to remove redundant filters during inference. We deploy IODF with TensorRT on GPUs, achieving $10\times$ inference speedup compared to the fastest existing neural compressors, while retaining the high compression rates on ImageNet32 and ImageNet64.