The signal level and shape of induced polarization responses are significantly affected by the current pulse duration and waveform. If not accounted for, this data dependency on the current will propagate trough the inversion to results rendering unquantifiable subsurface models. While this problem has been addressed in full-response induced polarization modelling, questions remain as to how to accurately retrieve quantitative induced polarization inversion models from the types of apparent integral chargeability data often used in data interpretation. Although several methodologies have been proposed for handling and inverting apparent resistivity and integral chargeability, these cannot compensate for the data dependency on the current waveform and pulse duration. This paper presents a novel inversion method for such data. The method considers current waveform and receiver transfer functions for retrieving quantitative IP models unbiased by transmitter waveform. The method uses the constant phase angle model, expressed in terms of the medium resistivity and phase. Specifically, four field data sets for the same profile but with different 100 per cent duty cycle pulse durations (4, 2, 1 and 0.5 s) serve as examples of data sets giving models dependant on current waveform when inverted with standard approaches. The novel inversion method presented here gives quantifiable models independent on the current waveform and pulse duration. These results resemble models retrieved with existing, full-response induced polarization inversions. The results still contain some degree of uncertainty in relation to underlying assumptions and parametrizations. Managing this source of uncertainty is considered in terms of full-response induced polarization inversions with constant phase angle and maximum phase angle inversions.