Actually, there's NO mention of any such mantras. It's a translation oversight.
- In fact, the "immortal crow", sage Kāka-Bhuśuṇḍi survives the Kalpānta-pralaya using the Aṣṭāṅga Yoga, specifically the 6th limb of the Yoga - Dhāraṇa (steady concentrated-abstraction of mind).
This question is a part of the 'Mistranslated Verses Series', the other two cases of which I discuss in my answer here and here, underlying the importance of referring to multiple translations in case of ambiguous verses.
Let's see the original Sanskrit verses of the cited English translation
Nirvāṇa Prakaraṇa - Pūrvārddha: Chapter 21
Verse 6.1.21.17
प्रतपन्ति यदादित्याः शकलीकृतभूधराः ।
वारुणीं धारणां बद्ध्वा तदा तिष्ठामि धीरधीः ॥ १७ ॥
pratapanti yadādityāḥ śakalīkṛtabhūdharāḥ |
vāruṇīṃ dhāraṇāṃ baddhvā tadā tiṣṭhāmi dhīradhīḥ |17|
Verse 6.1.21.18
यदा शकलिताद्रीन्द्रा वान्ति प्रलयवायवः ।
पार्वतीं धारणां
बद्ध्वा खे तिष्ठाम्यचलं तदा ॥ १८ ॥
yadā śakalitādrīndrā vānti pralayavāyavaḥ |
pārvatīṃ dhāraṇāṃ baddhvā khe tiṣṭhāmyacalaṃ tadā |18 |
Verse 6.1.21.19
जगद्गलितमेर्वादि यात्येकार्णवतां यदा ।
वायवीं धारणां बद्ध्वा
संप्लवेऽचलधीस्तदा ॥ १९ ॥
jagadgalitamervādi yātyekārṇavatāṃ yadā |
vāyavīṃ dhāraṇāṃ baddhvā
saṃplave'caladhīstadā |19|
Therefore, we can easily see and guess from the Sanskrit verses, even without the translation, that in the three verses, Bhuśuṇḍi talks about the three Dhāraṇas (and not any mantras) viz.
- The 'vāruṇī dhāraṇā' - concentration on the Water elements.
- The 'pārvatī dhāraṇā' - concentration on the (rock-hard mountains) Earth elements.
- The 'vāyavī dhāraṇā' - concentration on the Air Elements.
Corrected English Translations of the mistranslated verses.
Still in doubt?
Let me reproduce here the Hindi Translation of the above three verses by Dr. Mahāprabhulāla Gosvāmī, to attest the above claim:
Verse 6.1.21.17
प्रतपन्ति यदादित्याः शकलीकृतभूधराः ।
वारुणीं धारणां बद्ध्वा तदा तिष्ठामि धीरधीः ॥ १७ ॥
प्रलय के समय में जब पर्वतों को खण्डशः कर देने वाले बारह आदित्य तपते
हैं, तब मैं वरुण सम्बन्धी धारणा बाँध कर निश्चित बुद्धि होकर स्थित रहता
हूँ । जिस प्रकार जल के बीच रहने वाले वरुण बाहर के आतप से जनित संताप का
अनुभव नहीं करते, उसी प्रकार भुशुण्ड भी 'अत्यन्त शीतल समस्त दिग्-मण्डल
में व्यापी अपरिच्छिन्न जल स्परूप वरुण ही मैं हूँ' इस प्रकार चित्त में
निरन्तर वरुण-भावना के द्वारा वरुण रूप होकर बाहर के सूर्यादि-आतपजनित
संताप का अनुभव नहीं करते थे ॥१७॥
During the time of Pralaya, when the twelve sweltering Ādityas, which are
capable of fragmenting even the mountains, starts glowing, thereupon I resort to
the Dhāraṇā related to Vāruṇa with my Mind completely fixed upon the water
elements.
- The Translator notes:- Just like how the water-dweller Vāruṇa-deva doesn't experience the effects of the sweltering heat from the Sun, similarly, the Kāka resorting to the 'vāruṇī dhāraṇā', concentrates his complete Self on the Water-elements of the god (identifying oneself as non-different from the Extremely cool, the all-pervading and indeterminate Water principle), and thus stays untouched from the excruciating heat issuing from the Kalpānta-pralaya Suns.
Verse 6.1.21.18
यदा शकलिताद्रीन्द्रा वान्ति प्रलयवायवः ।
पार्वतीं धारणां बद्ध्वा खे तिष्ठाम्यचलं तदा ॥ १८ ॥
जब प्रलय काल में बड़े-बड़े पर्वतराजों को मर्दित कर देने वाले
प्रलयकालीन वायु बहते हैं, तब मैं पर्वत सम्बन्धिनी धारणा बाँध कर
आकाश-मण्डल में अचल होकर स्थित रहता हूँ । प्रलय काल में पृथ्वी पर स्थित
पर्वतों के विनाश होने के कारण उन्हीं पर प्रलयकालीन वायुओं का आघात होता
है, इसलिए प्रलयकालीन वायुओं से होने वाले आघात के अविषय ब्रह्माण्ड के
बहिर्भूत आकाश में साधारण वायु से भी क्षोभ न होने के लिए पर्वत
सम्बन्धिनी धारणा बाँधकर मैं स्थित रहता हूँ ॥१८॥
At the time of dissolution, when powerful winds capable of destroying even
the biggest of the mountains, start flowing, then I resort to the Dhāraṇā
related to the mountains, concentrating my mind completely, I firmly
establish me in the Space, becoming immovable.
- The Translator notes:- During the Dissolution period, as the great mountains get desecrated from the gargantuan winds, Bhuśuṇḍi situates himself beyond the imperceptible Open space, holding himself in the "immovable-mountain concentration (Dhāraṇā)" to avoid getting affected by the destructive winds.
Verse 6.1.21.19
जगद्गलितमेर्वादि यात्येकार्णवतां यदा ।
वायवीं धारणां बद्ध्वा संप्लवेऽचलधीस्तदा ॥ १९ ॥
जब प्रलय काल में मेरु आदि पर्वत के गल जाने से जगत् एक समुद्र स्वरूप हो
जाता है, तब वायु सम्बन्धिनी धारणा बाँध कर एकमात्र वायु में ही
तादात्म्य भाव से निश्चित-बुद्धि होकर ऊपर तैरता रहता हूँ ॥ १९ ॥
- Finally, as the dissolution reaches its zenith, and even the Mountain Meru has been dissolved in the cosmic waters, then I resort to the "Dhāraṇā related to the Air elements, situating my Self as a non-different entity from the Air principle.
All the 'Corrected' English Translation of the Hindi Translation by Self.
In the later chapters (23 to 27), the same things on the Yoga are explained by Bhuśuṇḍi, thereby, explaining his cause of longevity.
Conclusion:
Therefore, we conclude that the 'immortal bird' viz. Kākabhuśuṇḍi survives the Pralaya mainly using the sixth limb of the Patañjali's Aṣṭāṅga Yoga i.e., the Dhāraṇa via concentrating on the Water (Vāruṇa), Mountain (Pārvatī), and the Air (Vāyu) Principles one by one.
There's no mantra involved in this process, as wrongly translated by the English Translator cited in the question.
A description of the 'Principles of the Dhāraṇa on the Five Great-Elements' is explained in the Skanda-Purāṇa Verses 4.1.41 (112-118), but I'm not sure if Bhuśuṇḍi is talking about those only or not.