High-energy cosmic neutrinos are expected to be produced in extremely energetic astrophysical sources such as active galactic nuclei (AGNs) and gamma ray bursts (GRBs). The IceCube Neutrino Observatory has recently detected a diffuse astrophysical neutrino flux at 5.7σ significance. One of the outstanding questions regarding astrophysical neutrinos is their flavor composition. Most standard oscillation scenarios predict tau neutrinos (υ_τ) in the astrophysical flux, which have a negligible background from cosmic ray induced atmospheric neutrinos. This work reports a search for high-energy astrophysical υ_τ with the IceCube neutrino observatory. This analysis has devised a new method to search for astrophysical υ_τ in the IceCube waveforms, and it is the first υ_τ analysis in IceCube that is more sensitive to a υ_τ flux than other neutrino flavors. A total of three years of data was used, yielding a sensitivity of 5.1 x 10^-8 GeV cm^-2 s^-1 sr^-1 near the PeV region assuming a flux limit of E^2 Φ_υ < 1.0 x 10^-8 GeV cm^-2 s^-1 sr^-1 per neutrino flavor. Zero events were found. Differential upper limits for astrophysical υ_τ are derived in the Ο(100) TeV to Ο(10) PeV regime based on the zero findings.