  Massive stars are important building blocks of the Universe, and their
stellar structure and evolution models are fundamental cornerstones of various
fields in modern astrophysics. The precision of these models is limited by our
lack of understanding of various internal mixing processes that significantly
influence the lifetime of these objects (e.g. core overshoot, chemical mixing,
or the internal differential rotation). Our goal is to calibrate models by
extending the sample of available seismic studies of slowly pulsating B (SPB)
stars, providing input for theoretical modelling efforts that will deliver
precise constraints on the parameters describing the internal mixing processes
in these objects. We used spectral synthesis and disentangling techniques to
derive fundamental parameters and to determine precise orbital parameters from
high-resolution spectra. We employed custom masks to construct light curves
from the Kepler pixel data and used standard time-series analysis tools to
construct a set of significant frequencies for each target. These sets were
first filtered from combination frequencies, and then screened for period
spacing patterns. We detect gravity mode period series of modes of the same
degree l with consecutive radial order n in four new and one revisited SPB
star. These predominantly prograde dipole series (covering ~10-40 radial
orders) are influenced by rotation and carry signatures of chemical mixing
processes. Our spectroscopic analysis, in addition to placing each object
inside the SPB instability strip and identifying KIC 4930889 as an SB2 binary,
reveals that KIC 11971405 is a fast rotator that shows very weak Be signatures.
Together with the observed photometric outbursts this illustrates that this Be
star is a fast rotating SPB star. We hypothesise that the outbursts might be
connected to its very densely compressed oscillation spectrum.
