The recital of Tehillah L’David (Ashrei) and Kedushah d’Sidra is more important than saying Tachanun, for the Chachamim laud the person who recites Tehillah L’David three times daily, declaring that he has a place in the World to Come (Berachot 4b). About Kedushah d’Sidra, Chazal say that it is one of the things in whose merit the world exists after the destruction of the Temple (Sotah 49a), whereas the recital of Tachanun is voluntary (Tur 131, in the name of Rav Natrunai Gaon). Therefore it is better that an individual praying on his own, who does not have time to complete everything, recites Tehillah L’David and Kedushah d’Sidra (Yechaveh Da’at 2:8).
Additionally, the virtue in reciting Tehillah L’David and Kedushah d’Sidra, whose basis is derived from the Talmud, is preferable to the recital of Shir Shel Yom (the Psalm of the Day), Pitum HaKetoret (the passage of the incense), and Aleinu L’Shabe’ach, which were customarily recited during the time of the Geonim and Rishonim (Avnei Yashfeh 9:58-60).
One who is praying in a minyan, and must leave before the end of the prayer service, should try to stay at least until after Kedushah d’Sidra, and even refrain from beginning to remove his tefillin before the conclusion of its recital (Shulchan Aruch 25:13). When possible, he should leave his tefillin on until after Kaddish-Titkabal (Mishnah Berurah 25:56). All this is only in extenuating circumstances; however, l’chatchilah, one must hear another two Kaddishim – Kaddish Yehei Shelama after Shir Shel Yom and Kaddish d’Rabbanan after Pitum HaKetoret.