This letter presents the design and development of a novel compact motor-based non-linear stiffness soft actuator used for tendon-driven joints, i.e. a magnet-integrated soft actuator (MISA). By applying a pair of antagonistic MISAs to a joint, it is capable of achieving joint stiffness adjustment and joint torque adaptation. The joint with MISAs can achieve a wide range of joint stiffness, about 520% of that obtained by linear stiffness actuators, i.e. LSSA (share the same allowable elastic force and elastic travel with MISA). A stiffness range of 0-121.86 Nm and deflection of $\pm 8.9\mathrm{^{\circ }}$ can be achieved when MISAs are applied to a joint with a moment arm of 180 mm. The MISA achieves a high Power/Volume ratio of 345.02 $\times\, 10^{3}$ W/m$^{3}$. The application of the MISA in the development of a biomimetic robotic arm is also presented, with the investigation of joint stability and output torque analyse under various joint stiffnesses. Two experiments show that joints with MISAs can emulate the state of human joints when the muscles are tensed and relaxed. The dumbbell lifting test illustrates that the MISAs are capable of actuating the robot with desirable performance.