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c = \frac{1}{\rho} \frac{d\rho}{dp}  = \frac{d \ln \rho}{dp} =  \frac{d }{dp} \left(  \ln  \left(\frac{p}{Z} \right)  \right) = \frac{Z}{p} \cdot \frac{d }{dp} \left(\frac{p}{Z} \right) = \frac{Z}{p} \cdot \left( \frac{1}{Z} + p \cdot \frac{d }{dp} \left( \frac{1}{Z} \right)   \right) = \frac{1}{p}  - \frac{1}{Z} \frac{dZ}{dp}

\frac{d \ln Z}{dp} = \frac{1}{p} - c(p) \rightarrow \ln \frac{Z}{Z_0} = \ln \frac{p}{p_0} - \int_{p_0}^p c(p) \, dp