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An Optimum Integral Design is introduced for cosine(nθ) coils where the entire end-to-end length of the coil generates field with the dilution from ends practically eliminated. The benefits of such a design are particularly significant in short magnets where the overall coil length is comparable to or a few times the coil diameter. The integral field strength is further enhanced since the design allows a larger number of turns than in typical magnet coils. In this concept, the ends and body harmonics are optimized together to create an integral cosine(nθ) azimuthal current distribution. The concept was initially developed for wire/cable wound magnets where the bend radius of turns in the ends can be small. However, the benefit of this general approach can be applied to cable magnets as well. The magnetic design of a corrector dipole for the AGS helical magnet, which was recently built and tested, is presented as one of several examples. The other examples include a few sub-compact designs: a dipole with coil length less than a coil diameter, a quadrupole with coil length less than a coil radius, etc. Apart from generating a large integral field for the given length, the computed integral field harmonics in these designs are only a few parts in 10,000 at 2/3 of the coil radius.