Advancements in medical imaging and endovascular grafting have facilitated minimally invasive treatments for aortic diseases. Accurate 3D segmentation of the aorta and its branches is crucial for interventions, as inaccurate segmentation can lead to erroneous surgical planning and endograft construction. Previous methods simplified aortic segmentation as a binary image segmentation problem, overlooking the necessity of distinguishing between individual aortic branches. In this paper, we introduce Context Infused Swin-UNet (CIS-UNet), a deep learning model designed for multi-class segmentation of the aorta and thirteen aortic branches. Combining the strengths of Convolutional Neural Networks (CNNs) and Swin transformers, CIS-UNet adopts a hierarchical encoder-decoder structure comprising a CNN encoder, symmetric decoder, skip connections, and a novel Context-aware Shifted Window Self-Attention (CSW-SA) as the bottleneck block. Notably, CSW-SA introduces a unique utilization of the patch merging layer, distinct from conventional Swin transformers. It efficiently condenses the feature map, providing a global spatial context and enhancing performance when applied at the bottleneck layer, offering superior computational efficiency and segmentation accuracy compared to the Swin transformers. We trained our model on computed tomography (CT) scans from 44 patients and tested it on 15 patients. CIS-UNet outperformed the state-of-the-art SwinUNetR segmentation model, which is solely based on Swin transformers, by achieving a superior mean Dice coefficient of 0.713 compared to 0.697, and a mean surface distance of 2.78 mm compared to 3.39 mm. CIS-UNet's superior 3D aortic segmentation offers improved precision and optimization for planning endovascular treatments. Our dataset and code will be publicly available.