The integrated refrigerator composed of sorption cooler and adiabatic demagnetization refrigerator (ADR) is proposed for sub-kelvin cooling without vibration. In this configuration, the demagnetization process of the ADR is supposed to start after the termination of the sorption cooling. The liquid helium-4 inside the thermosiphon of the sorption cooler should evaporate as quickly as possible. This paper presents the results of preliminary test of the sorption cooling process. The developed sorption cooler primarily consists of the sorption pump filled with activated charcoal of 5.2 g and the helium thermosiphon. The adsorption capacity of the sorption pump is experimentally confirmed. Furthermore, the empirical correlation of adsorbate concentration which predicts the experimental results within 15% error in the temperature range of 5-70 K is proposed. The sorption pump is thermally connected to the 4.5 K heat sink through a pure tin strap, and regulates its temperature by the heater installed on it. Sorption cooling experiment is conducted under two experimental conditions. The experiment shows that the cooler can refrigerate the thermal capacitor to 2.8 K from 4.5 K heat sink, and provides the cooling capacity of 2.29 J. Moreover, it is confirmed that the pumping speed of the sorption pump is dominantly affected by its transient cool-down process. The new configuration of the thermosiphon condenser is proposed to enhance the evaporation characteristic of liquid helium while the pumping speed of the sorption pump is not changed. Since the new thermosiphon has smaller gas volume, liquid helium evaporates more quickly due to the rapid reduction of the vapor pressure inside the thermosiphon.