The West does have a very beautiful prayer that specifically invokes the Trinity. Unfortunately, it is seldom heard, except perhaps in the music of William Byrd who set the text to music in the 16th century. It's called Tribue Domine:
Tribue Domine, ut donec in hoc fragili corpore positus sum, laudet te cor meum, laudet te lingua mea, et omnia ossa mea dicant: Domine, quis similis tui? Tu es Deus omnipotens, quem trinum in personis, et unum in substantia deitatis colimus et adoramus, Patrem ingenitum, Filium de Patre unigenitum, Spiritum sanctum de utroque procedentem, et in utroque permanentem, Sanctam et individuam trinitatem, unum Deum omnipotentem.
Te deprecor, supplico et rogo, auge fidem, auge spem, auge charitatem. Fac nos per ipsam gratiam tuam semper in fide stabiles, et in opere efficaces, ut per fidem rectam et condigna fidei opera ad vitam te miserante, perveniamus æternam.
Gloria Patri qui creavit nos. Gloria Filio qui redemit nos. Gloria Spiritui sancto qui sanctificavit nos. Gloria summæ et individuæ trinitati cujus opera inseparabilia sunt, cujus imperium sine fine manet. Te decet laus, te decet hymnus, tibi debetur omnis honor, tibi benedictio et claritas, tibi gratiarum actio, tibi honor, virtus et fortitudo, Deo nostro, in sæcula sæculorum. Amen.
Grant, O Lord, that while I am placed in this feeble body my heart shall praise Thee, my tongue shall praise thee, and all my bones shall say: Lord, who is like unto Thee? Thou art God Almighty, Who as three persons in one divine substance we worship and adore: Father unbegotten, only-begotten Son of the Father, Holy Spirit proceeding from both and remaining in both, Holy and undivided Trinity, one God Almighty.
I entreat, pray and request Thee, increase my faith, increase my hope, increase my charity: make us through Thy grace forever steadfast in faith and effective in works, that through true faith and works worthy of our faith we may come through, by Thy mercy, to eternal life.
Glory be to the Father, who created us: glory be to the Son, who saved us, glory be to the Holy Spirit, who sanctified us: glory be to the highest and undivided Trinity, whose works are inseparable, whose reign remains without end. Praise and hymn become Thee, to Thee are due all honour, blessing and glory, to Thee be thanksgiving, honour, power and might, our God, world without end. Amen.