The prayer recited by Yudhishtira is from the beginning chapters of Virata Parva. It can be found in Chapter 6 of Virata Parva. This is famous as Yudhiṣṭira kṛta Durgā stavam or Yudhiṣṭira kṛta Durgā stotram. The prayer consists of 36 verses. This was narrated to Janamejaya by Vaishampayana.
Yudhishtira sung this prayer to Durga while he was on the way to Virata kingdom to spend his one year exile. The text is as follows:
virāṭanagaraṃ ramyaṃ gacchamāno yudhiṣṭhiraḥ ।
astuvanmanasā devīṃ durgāṃ tribhuvaneśvarīm ॥1॥
yaśodāgarbhasaṃbhūtāṃ nārāyaṇavarapriyām ।
nandagopakule jātāṃ maṅgalyāṃ kulavardhanīm ॥2 ॥
kaṃsavidrāvaṇakarīmasurāṇāṃ kṣayaṃkarīm ।
śilātaṭavinikṣiptāmākāśaṃ prati gāminīm ॥ 3 ॥
vāsudevasya bhaginīṃ divyamālyavibhūṣitām ।
divyāmbaradharāṃ devīṃ khaṅgakheṭakadhāriṇīm ॥ 4 ॥
bhārāvataraṇe puṇye ye smaranti sadā śivām ।
tānvai tārayate pāpātpaṅke gāmiva durbalām ॥ 5 ॥
stotuṃ pracakrame bhūyo vividhaiḥ stotrasaṃbhavaiḥ ।
āmantrya darśanākāṅkṣī rājā devīṃ sahānujaḥ ॥ 6 ॥
namostu varade kṛṣṇe kumāri brahmacāriṇi ।
bālārkasadṛśākāre pūrṇacandranibhānane ॥ 7 ॥
caturbhuje caturvakre pīnaśroṇipayodhare ।
mayūrapicchavalaye keyūrāṅgadadhāriṇi ॥ 8 ॥
bhāsi devi yathā padmā nārāyaṇaparigrahaḥ ।
svarūpaṃ brahmacaryaṃ ca viśadaṃ tava khecari ॥ 9 ॥
kṛṣṇacchavisamā kṛṣṇā saṃkarṣaṇasamānanā ।
bibhratī vipulau bāhu śakradhvajasamucchrayau ॥ 10 ॥
pātrī ca paṅkajī ghaṇṭī strī viśuddhā ca yā bhuvi ।
pāśaṃ dhanurmahācakraṃ vividhānyāyudhāni ca ॥ 11 ॥
kuṇḍalābhyāṃ supūrṇābhyāṃ karṇābhyāṃ ca vibhūṣitā ।
candravispardhinā devi mukhena tvaṃ virājase ॥ 12 ॥
mukuṭena vicitreṇa keśabandhena śobhinā ।
bhujaṅgābhogavāsena śroṇisūtreṇa rājatā ॥ 13 ॥
vibhrājase cā''baddhena bhogeneveha mandaraḥ ।
dhvajena śikhipicchānāmucchritena virājase ॥ 14 ॥
kaumāraṃ vratamāsthāya tridivaṃ pāvitaṃ tvayā ।
tena tvaṃ stūyase devi tridaśaiḥ pūjyase'pi ca ॥ 15 ॥
trailokyarakṣaṇārthāya mahiṣāsuranāśini ।
prasannā me suraśreṣṭhe dayāṃ kuru śivā bhava ॥ 16 ॥
jayā tvaṃ vijayā caiva saṃgrāme ca jayapradā ।
mamāpi vijayaṃ dehi varadā tvaṃ ca sāṃpratam ॥ 17 ॥
vindhye caiva nagaśreṣṭhe tava sthānaṃ hi śāśvatam ।
kāli kāli mahākāli śīdhumāṃsapaśupriye ॥ 18 ॥
kṛtānuyātrā bhūtaistvaṃ varadā kāmacāriṇī ।
bhārāvatāre ye ca tvāṃ saṃsmariṣyanti mānavāḥ ॥ 19 ॥
praṇamanti ca ye tvāṃ hi prabhāte tu narā bhuvi ।
na teṣāṃ durlabhaṃ kiṃcitputrato dhanatopi vā ॥ 20 ॥
durgāttārayase durge tattvaṃ durgā smṛtā janaiḥ ।
kāntāreṣvavasannānāṃ magrānāṃ ca mahārṇave ।
dasyubhirvā niruddhānāṃ tvaṃ gatiḥ paramā nṛṇām ॥ 21 ॥
jalaprataraṇe caiva kāntāreṣvaṭavīṣu ca ।
ye smaranti mahādevi na ca sīdanti te narāḥ ॥ 22 ॥
tvaṃ kīrtiḥ śrīrdhṛtiḥ siddhirhrīrvidyā saṃtatirmatiḥ ।
saṃdhyā rātriḥ prabhā nidrā jyotsnā kāṃtiḥkṣamādayā ॥ 23 ॥
nṛṇāṃ ca bandhanaṃ mohaṃ putranāśaṃ dhanakṣayam ।
vyādhiṃ mṛtyuṃ bhayaṃ caiva pūjitā nāśayiṣyasi ॥ 24 ॥
sohaṃ rājyātparibhraṣṭaḥ śaraṇaṃ tvāṃ prapannavān ।
praṇataśca yathā mūrdhnā tava devi sureśvari ॥ 25 ॥
trāhi māṃ padmapatrākṣi satye satyā bhavasva naḥ ।
śaraṇaṃ bhava me durge śaraṇye bhaktavatsale ॥ 26 ॥
evaṃ stutā hi sā devī darśayāmāsa pāṇḍavam ।
upagamya tu rājānamidaṃ vacanamabravīt ॥ 27 ॥
This is how Yudhishtira praised goddess Durga. Now, goddess replied thus:
Devyuvāca ।
śṛṇu rājanmahābāho madīyaṃ vacanaṃ prabho ।
bhaviṣyatyacirādeva saṃgrāme vijayastava ॥ 28 ॥
mama prasādānnirjitya hatvā kauravavāhinīm ।
rājyaṃ niṣkaṇṭakaṃ kṛtvā bhokṣyase medinīṃ punaḥ ॥ 29 ॥
bhrātṛbhiḥ sahito rājanprītiṃ prāpsyasi puṣkalām ।
matprasādācca te saukhyamārogyaṃ ca bhaviṣyati ॥ 30 ॥
ye ca saṃkīrtayiṣyanti loke vigatakalmaṣāḥ ।
teṣāṃ tuṣṭā pradāsyāmi rājyamāyurvapuḥ sutam ॥ 31 ॥
pravāse nagare vā'pi saṃgrāme śatrusaṃkaṭe ।
aṭavyāṃ durgakāntāre sāgare gahane girau ॥ 32 ॥
ye smariṣyanti māṃ rājanyathā'haṃ bhavatā smṛtā ।
na teṣāṃ durlabhaṃ kiṃcitasmim̐lloke bhaviṣyati ॥ 33 ॥
idaṃ stotravaraṃ bhaktyā śṛṇuyādvā paṭheta vā ।
tasya sarvāṇi kāryāṇi siddhiṃ yāsyanti pāṇḍavāḥ ॥ 34 ॥
matprasādācca vaḥ sarvānvirāṭanagare sthitān ।
na prajñāsyanti kurako narā vā tannivāsinaḥ ॥ 35 ॥
ityuktvā varadā devī yudhiṣṭhiramariṃdamam ।
rakṣāṃ kṛtvā ca pāṇḍūnāṃ tatraivāntaradhīyata ॥ 36 ॥ ॥
Boons given by goddess:
Goddess Durga was satisfied by the prayer of Yudhishtira and assured him the victory over the Kauravas. She assured him the lordship over the worlds and his lost kingdom back. She bestowed her grace over Yudhishtira. She gave boon of happiness, joy and wealth to the Pandavas.
The result (phala shruti) of praying Durga with his hymn and singing her attributes:
I will bestow upon them kingdom, long life, beauty of person, and offspring. And they, O king, who will invoke me, after thy manner, in exile or in the city, in the midst of battle or of dangers from foes, in forests or in inaccessible deserts, in seas or mountain fastnesses, there is nothing that they will not obtain in this world.
The goal of the Pandavas at that moment of time was completing the exile in the Matsya (name of Virata's kingdom) kingdom without being recognized by anyone. She gives the boon of completion of the exile successfully. Having said this, the goddess disappeared there and then.