The integration of Artificial Intelligence (AI) into Integrated Development Environments (IDEs) is reshaping software development, fundamentally altering how developers interact with their tools. This shift marks the emergence of Human-AI Experience in Integrated Development Environment (in-IDE HAX), a field that explores the evolving dynamics of Human-Computer Interaction in AI-assisted coding environments. Despite rapid adoption, research on in-IDE HAX remains fragmented, which highlights the need for a unified overview of current practices, challenges, and opportunities. To provide a structured overview of existing research, we conduct a systematic literature review of 90 studies, summarizing current findings and outlining areas for further investigation. We organize key insights from reviewed studies into three aspects: Impact, Design, and Quality of AI-based systems inside IDEs. Impact findings show that AI-assisted coding enhances developer productivity but also introduces challenges, such as verification overhead and over-reliance. Design studies show that effective interfaces surface context, provide explanations and transparency of suggestion, and support user control. Quality studies document risks in correctness, maintainability, and security. For future research, priorities include productivity studies, design of assistance, and audit of AI-generated code. The agenda calls for larger and longer evaluations, stronger audit and verification assets, broader coverage across the software life cycle, and adaptive assistance under user control.