One can recite out loud or silently during the Tahajjud prayer

Abd Allah bin Abu Qais said:

I asked 'Aishah about the witr observed by the Messenger of Allah (sallallaahu alayhi wa sallam). She replied: Sometime he observed the witr prayer in the early hours of the night, sometimes he observed it at the end of it. I asked: How did he recite the Qur'an ? Did he recite the Qur'an quietly or loudly?

She replied: He did it in any way. Sometimes he recited quietly and sometimes loudly, sometimes he took bath and then slept and sometimes he performed ablution and then slept.

Abu Dawud said: The narrators other than Qutaibah said: This refer to his bath due to sexual defilement.

Sunan Abi Dāwud Hadīth No. 1437

وفي الحديثِ: الحِرصُ على مَعرفةِ سُننِ النَّبيِّ صلَّى اللهُ عليه وسلَّم.

The Hadīth shows us how the Companions had a love and desire to learn the Sunan of the Prophet sallallaahu alayhi wa sallam

وفيه: بيانُ أنَّ الدِّينَ يُسرٌ لا عسرٌ، وقد بيَّن ذلك النَّبيُّ صلَّى الله عليه وسلَّم بهَدْيِه وسُنَّتِه

The Hadīth also shows us that the Dīn is ease and not hardship. The Messenger of Allāh sallallaahu alayhi wa sallam made that clear by means of his guidance and his Sunnah.