What is the origin of Naam Traya Mantra? Or in which scripture this mantra is mentioned? The mantra is Om Achutaya namaha Anantaya namaha Govindaya Namaha. I don't know the correct form but these 3 names of God Vishnu are there in Naam Traya Mantra. I also got the Rishi and Chanda from internet but don't know in which scripture this information is found?
In the 19th patala of Aanadikalpa-Tantra
ऋषय उचुः- नामत्रयस्य माहात्म्यं श्रोतुकामा वयं प्रभो। भगवन् श्रोतुमिच्छामो वक्तुमर्हस्यशेषत:।
Rishis said: O Prabho, We wish to here about the Nama-Traya Mahatmya! Please enlighten us with it.
ब्रह्मोवाच अतिगुह्यतरं मन्त्र वर्षकोटिशतैरपि। नशक्यं विस्तराद वक्तुं संक्षेपाच्छूणुत द्विजा:॥ १ ॥
What you wish to here is highly secret. Even if I speak for a hundred crore years, I would be unable to describe it completely. I shall speak briefly,
अथ अच्युतादों पूजादिः अच्युतानन्तगोविन्दैश्चतुर्थयन्तैस्त्रिभि: पदेः । नमोऽन्तैर्जपकाले तु मूलमन्त्र इतीरितः। अथवापि समस्तैस्तु चतुर्थ्यन्तैकमन्त्रता ।
know the three mantras, Achutaya Namaha, Anantaya Namaha and Govindaya Namaha to be three distinct Mool-Mantras (अच्युताय नमः, अनन्ताय नमः, गोविन्दाय नमः). One must however use Achuta-Nanta-Govindaya Namaha (अच्युतानन्तगोविन्दाय नमः ) in japa and puja.
तदृषि: शौनको ज्ञेयः पृथक् पक्षे पराशर:। व्यासश्च नारदश्चैव विराट्छन्द उदीरितः। परं ब्रह्म तथा प्रोक्तं हरिर्वा देवतेत्यपि ।
The seer of Achuta-Nanta-Govindaya Namaha is Shaunaka, and that of the Mool-mantras is Parashara, Vyasa and Narada respectively. All the four mantras are in Virat Chandas (विराट्). The devta is Hari.
So going by the above, the Seers match with the one in the question. The Dhyanam given in Aanadikalpa is same as in the question.
These Mantras are found in the Padma Purana and are also used during Achamana by certain sects.
Quoting from the book "Essence of Padma Purana" @Kamakoti.org:
The arrangement of the ‘Samudra Mathana’was that Manthara Mountain was to be set as the churning rod, Maha Sarpa Vasuki was the string with which to tie up the Mountain and Bhagavan himself as balancing Shakti as the Avatar of Kurma (Tortoise) to ensure that the Meru Mountain would be made stable and steady. As the churning started, innumerable of the proud Danavas who opted for the head of thousand faced Vasuki out of pride got perished due to the poisonous flames where as Devas were at the tail end. Eventally, a massive ‘Halahala’ or the sky-rocketing poisonous flames engulfed the Univerese and by the grace of Three Names of Lord Vishnu viz. Om Achyutaya Namah, Om Anantaya Namah and Om Govindaya Namah, Bhagavan Shankara gulped the blue poisonous flames in his throat thus giving him the epithet of Neela Kantha. By the power of the three names of Vishnu, there would never be any hardship faced:
Achyutaananta Govinda iti naamatrayam Maha Mantram hareh/
yo japetpriyato Bhaktaayaa Pranavaadhyam namontakam/
Tasya Mrityu bhayam naasti visharogaagnijam mahat/
Naamatrayam Maha Mantramjapedyam prayatatmavaan/
Kaala Mrityu bhayam chaapi tasya naasti kimanyatah
Whoever recites the Three Names viz. Om Achyutaaya, Om Anantaaya, and Om Govindaaya sincerely and with dedication, none of the poisonous diseases or Agni- based frights/deaths would affect.
Here, as one can see, the context is Samudra Manthana.
But it is also quite possible that the Mantras are also found in some other scriptures as well.
I have found the Rishi-Devata etc for the Nama Traya Mantra in Nityotsava's 3rd Chapter, the Shri Krama. But the they are not matching with the ones you have given. I am giving below the limbs as well as the Dhyanam for the Mantra:
NAmatraya mantrasya kAshyapAtribharadvAjA rishayah | Anushtup Chandah | ShrimahAvishnur devatA | Tat prityarthe jape viniyogah ||
Samastadurastar vyAdhi sangha dhvamsa patiyase |
Achyut ananta govinda nAmne dhAmne namo namaha ||
Note that there are three Rishis - Kasyapa, Atri and BharadvAja - mentioned as the seers of the Mantra.
As per PADMA-PURANA: UTTARAKHANDA: Chapter 78 The Hymn called 'Apamarjana',
1-2. I shall hereafter tell you about the excellent vow of the sprinkling with water, as told by Pulastya to the magnanimous Dalabhya. It destroys all faults caused by diseases. It gives auspiciousness. I shall tell it to you. Listen, O daughter of the (Himalaya) Mountain.
Sri Pulastya said [to Sri Dalabhya]:
45-65. ... By means of the medicine of the utterance of the names, viz. Acyuta, Ananta, Govinda, all the diseases vanish. ...
It's references can be found during Samudra Manthan episode also.
PADMA-PURANA: BHUMIKHANDA: Chapter 9. The Churning Starts:
2-7a. Seeing them running away Shankara said these (words): "O hosts of gods, hand over the poison to me. I shall quickly inactivate the great (i.e. severe) poison, viz. Kalakuta". > Saying so, that lord of Parvati, meditating upon Visnu in his heart and uttering the great hymn, took that fearful poison. By the power of the great hymn, that great (deadly) poison was digested.
7b-22. 7b-22. He who, being controlled, would devoutly mutter the three names, viz. Acyuta, Ananta, and Govinda, of Visnu with Pranava (i.e. Om) at the beginning and Om at the end, has no fear of taking poison, or from fire and also from death. ...
PADMA-PURANA: UTTARKHANDA: Chapter 232. The Raise of Goddess Laksmi:
16-21. Having meditated (i.e. when I meditated) with a concentrated mind upon that lord along with J§rl and Bhumi, having earrings of gold purified by fire, remover of all miseries, and on my having muttered the great hymn of the name and form (of Visnu) along with Mahalaksmi, all that poison, which was very fierce, which was the first one, which was fearful to all, which was (capable of) destroying all the worlds, was digested (by me) due to the three names of Visnu, the omnipresent one. He who, being restrained, would devoutly mutter the three names of Visnu, viz. Acyuta, Ananta, Govinda, beginning with Pranava (i.e. Om) and (also) ending with Om, has no fear of death, so also the great fear due to poison, disease and fire. The wise, restrained one, who would mutter the great hymn—the three names—does not have fear of Death; then from where else (can he have it)? Thus with (the muttering of) the three names I drank that poison.
Okay since already other shastra pramanas are given, I will answer the question you asked in comments.
Can you try to find the Lalita Devi's incident of throwing Naam Traya Mantra Astra on Bhanadasur?
It appears in Lalitopakhyana,
The Chapter number varies as per different publications, but is put under the title of श्रीललिताभण्डासुरसंग्राम or श्रीललिताभण्डासुरदिव्यास्त्रप्रयोगह् ।
The Dānava discharged the missile of Mahārogas (great diseases) among the armies of Śaktis. Thereupon pulmonary consumption and thousands of other diseases began to ravage.
तन्निवारणसिध्यर्थम् ललिता परमेश्वरी।
नामत्रयमहामन्त्रम् महास्त्रम् संमुमोचह।।
In order to achieve their eradication and prevention, Lalitā, the great goddess, discharged the great missile of the great Mantra of the three names (of Lord Viṣṇu).
Acyuta, Ananta and Govinda rose up from the missile. By their Huṃkāra alone they burned those diseases and gave (relief and) delight, (to the Śaktis).
नत्वा च ताम् महेशानीं तभ्दक्तव्याधिमर्दनम्।
विधातुम् त्रिपु लोकेषु नियुक्ताह् स्वं पदम् ययुह्।।
They bowed down to Maheśānī. They were employed (by her) to suppress sickness of her devotees in all the three worlds. They went to their own region.