Computer-aided brain tumor segmentation using magnetic resonance imaging (MRI) is of great significance for the clinical diagnosis and treatment of patients. Recently, U-Net has received widespread attention as a milestone in automatic brain tumor segmentation. Following its merits and motivated by the success of the attention mechanism, this work proposed a novel mixed attention U-Net model, i.e., MAU-Net, which integrated the spatial-channel attention and self-attention into a single U-Net architecture for MRI brain tumor segmentation. Specifically, MAU-Net embeds Shuffle Attention using spatial-channel attention after each convolutional block in the encoder stage to enhance local details of brain tumor images. Meanwhile, considering the superior capability of self-attention in modeling long-distance dependencies, an enhanced Transformer module is introduced at the bottleneck to improve the interactive learning ability of global information of brain tumor images. MAU-Net achieves enhancing tumor, whole tumor and tumor core segmentation Dice values of 77.88/77.47, 90.15/90.00 and 81.09/81.63% on the brain tumor segmentation (BraTS) 2019/2020 validation datasets, and it outperforms the baseline by 1.15 and 0.93% on average, respectively. Besides, MAU-Net also demonstrates good competitiveness compared with representative methods.