In this paper we present undulator magnet tapering methods for obtaining high efficiency and multiterawatt peak powers in x-ray free electron lasers (XFELs), a key requirement for enabling 3D atomic resolution single molecule imaging and nonlinear x-ray science. The peak power and efficiency of tapered XFELs is sensitive to time dependent effects, like synchrotron sideband growth. To analyze this dependence in detail we perform a comparative numerical optimization for the undulator magnetic field tapering profile including and intentionally disabling these effects. We show that the solution for the magnetic field taper profile obtained from time independent optimization does not yield the highest extraction efficiency when time dependent effects are included. Our comparative optimization is performed
for a novel undulator designed specifically to obtain TW power x-ray pulses in the shortest distance: superconducting, helical, with short period and built-in strong focusing. This design reduces the length of the breaks between modules, decreasing diffraction effects, and allows using a stronger transverse electron focusing. Both effects reduce the gain length and the overall undulator length. We determine that after a fully time dependent optimization of a 100 m long Linac coherent light source-like XFEL we can obtain a maximum efficiency of 7%, corresponding to 3.7 TW peak radiation power. Possible methods to suppress the synchrotron sidebands, and further enhance the FEL peak power, up to about 6 TW by increasing the seed power and reducing the electron beam energy spread, are also discussed.