Soft robotics represents a new set of technologies aimed at operating in natural environments, including near, or inside the human body. To move within and interact with their environment, soft robots require artificial muscles to actuate movement. These artificial muscles need to be as strong, fast, and robust as their natural counterparts. Dielectric elastomer actuators (DEAs) constitute a highly promising class of materials, but typically exhibit low output forces and low energy densities, when used without rigid supports. Here we report a soft composite material made of strain stiffening elastomers and carbon nanotube electrodes, which actuates under an applied electric field and demonstrates a peak energy density of 19.8 J/kg. The result is close to the upper limit for natural muscle (40 J/kg) making these DEAs the highest performance electrical driven artificial muscles. To obtain high forces and displacements, we used low density, ultra-thin carbon nanotube electrodes which can sustain applied electric fields upwards of 100 V/micron, without suffering from dielectric breakdown. The fabrication process described herein is fast, scalable, and uses relatively low cost components and equipment. Potential future applications include biomedical uses, such as prosthetics, surgical robots, and wearable devices, as well as more capable soft robots capable of locomotion and manipulation in natural or human-centric environments.