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We compute the intrinsic isotropic peak luminosity function (LF) andformation rate of long gamma-ray bursts (LGRBs) using a novel approach. Wecomplement a standard log\,$N$\,--\,log\,$P$ brightness distribution and$V_{\mathrm{max}}$ estimations with two observation-time relations: aredshift--observation-time relation (log\,$z$\,--\,log\,$T$) and a newluminosity--observation-time relation (log\,$L$\,--\,log\,$T$). We show thatthis approach reduces degeneracies that exist between the rate and LF of abrightness distribution. To account for the complex triggering algorithmemployed by \emph{Swift} we use recent results of \citet{Lien_2014ApJ} toproduce a suite of efficiency functions. Using these functions with the abovemethods, we show that a log\,$L$\,--\,log\,$T$ method can provide goodconstraints on the form of the LF, particularly the high end. Using a sample of175 peak luminosities determined from redshifts with well defined selectioncriteria our results suggest that LGRBs occur at a local rate (without beamingcorrections) of $[\,0.7 < \rho_{0} <0.8\,]\,\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$. Within this range, assuming abroken-power-law LF, we find best estimates for the low and high energy indicesof $-0.95 \pm 0.09$ and $-2.59 \pm0.93$ respectively, separated by a breakluminosity $0.80 \pm0.43 \times 10^{52}$\,erg\,s$^{-1}$.

Constraining the rate and luminosity function of Swift gamma-ray bursts