  We study high efficiency, multi-terawatt peak power, few angstrom wavelength,
X-ray Free Electron Lasers (X-ray FELs). To obtain these characteristics we
consider an optimized undulator design: 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. The peak power and efficiency depend on the
transverse electron beam distribution and on time dependent effects, like
synchrotron sideband growth. The last effect is identified as the main cause
for reduction of electron beam microbunching and FEL peak power. We show that
the optimal functional form for the undulator magnetic field tapering profile,
yielding the maximum output power, depends significantly on these effects. The
output power achieved when neglecting time dependent effects for an LCLS-like
X-ray FEL with a 100 m long tapered undulator is 7.3 TW, a 14 $\%$ electron
beam energy extraction efficiency. When these effects are included the highest
peak power is achieved reducing the tapering rate, thus minimizing the
reduction in electron micro-bunching due to synchrotron sideband growth. The
maximum efficiency obtained for this case is 9 $\%$, corresponding to 4.7 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, are discussed.
