Are the
isti’aadhah and basmalah part of the opening du’aa’ of the prayer?
Question

                Is the ta'wuth and basmalah (seeking refuge
      in the name of Allah from the Shaitan and saying bismillah), are they part of the opening
      du'a? So If someone is praying behind the Imam should he say them quietly? Should he
      repeat the Bismillah in every raka'ah quietly if he is praying behind the Imam?
Praise be to Allah.The istiaadaha and basmalah are not part of the opening
    duaa reported from the Prophet 
 (peace and blessings of Allaah be upon
    him), although they are part of what should be recited at the beginning of the prayer.
    Aaishah (may Allaah be pleased with her) said: The Messenger of Allaah
    
 (peace and blessings of Allaah be upon him) used to open his prayer with takbeer
    (Allaahu akbar). (Reported by Muslim, no. 498)
    As regards saying the istiaadhah and basmalah in
    the second rakah, the scholars differ on this matter. The basmalah should be recited
    in every rakah, when reciting al-Faatihah, but it should not be recited
    aloud, because there is some dispute as to whether it is part of al-Faatihah or not. Those
    scholars who say that it is not part of al-Faatihah recommend that it should be recited
    before al-Faatihah in each rakah.
So to be on the safe side, the worshipper should recite the basmalah,
    silently, in every rakah.
With regard to the istiaadhah, there is no difference among the
    scholars, all of whom say that it is not part of Soorat al-Faatihah, but there is some
    discussion among them concerning the aayah (interpretation of the meaning): So
    when you want to recite the Quran, seek refuge with Allaah [al-Nahl
    16:98]
Those who take the aayah at face value say that the istiaadhah
    should be recited in every rakah, before reciting al-Faatihah and the basmalah. This
    is the opinion of the Shaafais and is the most correct opinion in their view. (Al-Majmoo,
    3/323). Al-Hasan al-Basri, Ataa and Ibraaheem al-Nakhai
    also recommended taawwudh in every rakah.
    Ibn Hazm (may Allaah have mercy on him) favoured this opinion, and it
    was also reported that Imaam Muhammad ibn Abd al-Wahhaab (may Allaah have mercy on
    him) mentioned it in Aadaab al-Mashiy ilal-Salaah. This opinion was also
    favoured by Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) (al-Insaaf
    2/74), who said in al-Ikhtiyaaraat: Taawwudh is
    recommended at the beginning of every recitation. (p.50). This is also the opinion
    of al-Allaamah al-Albaani.
    Some scholars say that it is to be recited at the beginning of the
    first rakah only, and does not have to be repeated in the other rakahs. This
    is the opinion of the Hanbalis (al-Insaaf, 2/73).
    Al-Shawkaani thought that the correct opinion was to recite it in the first rakah
    only. (Nayl al-Awtaar, 3/39, 139-140).
    The Prophet 
 (peace and blessings of Allaah be upon him) used to
    seek refuge with Allaah when he prayed. He used to say Aoodhu Billaahi min
    al-shaytaan il-rajeem min hamzihi wa nafkhihi wa nafathih (I seek refuge with Allaah
    from the accursed Shaytaan, from his evil insinuations). (reported by Abu
    Dawood). Sometimes he would expand on this and say, Aoodhu
    Billaahi il-Samee il-Aleem min al-shaytaan il-rajeem min hamzihi wa
    nafkhihi wa nafathih (I seek refuge with Allaah, the All-Hearing, the All-Seeing, from
    the accursed Shaytaan, from his evil insinuations). (Reported by Abu Dawood
    and al-Tirmidhi). And Allaah knows best.