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In the supplications to be recited after eyery mandatory prayer, the best is the Tasbeeh al-Fatima (as). It requires reciting:

اَللهُ أكْبَرُ

Allaho Akbar, Allah is the Greatest (34 times)
اَلحَمْدُ للهِ

Alhamdu lillah, Praise is for Allah (33 times)
سُبْحانَ اللهِ

Subhan Allah, Glory is to Allah (33 times)
Imam Muhammad Al-Baqir (as) has said that after every mandatory prayer, there is no better invocation than the Tasbeeh al-Fatima (as). If there was any other, more effective, way of Praising Allah, the Prophet (S) would most certainly haveinstructed his daughter about that.

Imam Ja’far As-Sadiq has said that reciting the Tasbeeh al-Fatima (as) after every compulsory prayer is better than performing a thousand genuflections. One should say once after reciting the Tasbeeh the following words:

لاَ إلَهَ إلاّ اللهُ

La ilaha il Allah, There is no God save He
Imam Muhammad Al-Baqir (as) has said, One who sits quietly after each mandatory prayer and recites thrice the following words, Allah will pardon all his sins:

أسْتَغْفِرُ اللهَ الّذِي لاَ إلَهَ إلاَّ هُوَ الحَيُّ القَيُّومُ ذُو الجَلالِ وَالإكْرَامِ وَأتُوبُ إلَيْهِ.

“I seek forgiveness from the Lord (testifying) There is no God other than Him The Living and the Everlasting Of Majesty and Splendour And I turn to Him In Repentence!”

Imam Ja’far As-Sadiq has said that if a person recites thirty times the following words after each compulsory prayer, all his sins will be pardoned:

سُبْحانَ اللهِ

Subhan Allah, Glory is to Allah
It is narrated from Amir’ul-Mu’mineen (as) if one desires to depart from this world as innocent as he was at his birth, then he should recite the Sura al-Iqlaas twelve times after every compulsory prayer and then raise his hands towards the sky and recite the following invocation:

اَللّهُمَّ إنِّي أسْأَلُكَ بِاسْمِكَ المَكْنُونِ المَخْزُونِ الطّاهِرِ الطُّهْرِ المُبَارَكِ، وَأسْأَلُكَ بِاسْمِكَ العَظِيمِ وَسُلْطَانِكَ القَدِيمِ، يا وَاهِبَ العَطايَا وَيَا مُطْلِقَ الأسَارَى وَيَا فَكَّاكَ الرِّقَابِ مِنَ النّارِ، أَسْأَلُكَ أنْ تُصَلِّيَ عَلَى مُحَمَّدٍ وَآلِ مُحَمَّدٍ وَأنْ تَعْتِقَ رَقَبَتِي مِنَ النّارِ، وَأنْ تُخْرِجَنِي مِنَ الدُّنْيا آمِناً، وأنْ تُدْخِلَنِيَ الجَنَّةَ سَالِماً، وأنْ تَجْعَلَ دُعائِي أوَّلَهُ فَلاحاً وَأوْسَطَهُ نَجاحاً وَآخِرَهُ صَلاحاً إنَّكَ أنْتَ عَلاَّمُ الغُيُوبِ.

“O Allah, I ask You by Your hidden, safeguarded, pure, purifying and blessed name, and I ask You by Your great name and Your everexisting power, O Giver of gifts and unleasher of captives, who unties necks from the hellfire, I ask you to send blessings on Mohammad and the children of Mohammad, and to free my neck from the hellfire, and to save my faith in this world and to let me into paradise, and to help me ask for wishes that start with prosperity, and continue in success, and end with goodness. إِنَّكَ أَنْتَ عَلامُ الغُيُوبِ ). You are indeed the Knower of the Unseen”
Imam Ar-Ridha’ (as) was asked about the method of offering salutations to the Prophet (S) and his Ahl-ul-Bayt (as) after the mandatory prayers. The Imam (as) asked one to say the following:

اَلسَّلامُ عَلَيْكَ يَا رَسُولَ اللهِ وَرَحْمَةُ اللهِ وَبَرَكَاتُهُ، اَلسَّلامُ عَلَيْكَ يَا مُحَمَّدُ بْنَ عَبْدِاللهِ، اَلسَّلامُ عَلَيْكَ يَا خِيَرَةَ اللهِ، اَلسَّلامُ عَلَيْكَ يَا حَبِيبَ اللهِ، اَلسَّلامُ عَلَيْكَ يَا صفْوَةَ اللهِ، اَلسَّلامُ عَلَيْكَ يَا أمِينَ اللهِ، أشْهَدُ أنَّكَ رَسُولُ اللهِ، وَأشْهَدُ أنَّكَ مُحَمَّدُ بْنُ عَبْدِاللهِ، وَأشْهَدُ أنَّكَ قَدْ نَصَحْتَ لأُمَّتِكَ، وَجَاهَدْتَ فِي سَبِيلِ رَبِّكَ، وَعَبَدْتَهُ حَتّى أتَاكَ اليَقِينُ، فَجَزاكَ اللهُ يَا رَسُولَ اللهِ أفْضَلَ مَا جَزَى نَبِيّاً عَنْ أُمَّتِهِ، اَللَّهُمَّ صَلِّ عَلَى مُحَمَّدٍ وَآلِ مُحَمَّدٍ أَفْضَلَ مَا صَلَّيْتَ عَلَى إبْرَاهِيمَ وَآلِ إبْرَاهِيمَ إنَّكَ حَمِيدٌ مَجِيدٌ.

“Peace be upon you O Messenger of Allah and His mercy and blessings be upon you, Peace be upon you O Mohammad son of Abdullah, Peace be upon you O Elect of Allah, Peace be upon you O Beloved of Allah, Peace be upon you O Preferred by Allah, Peace be upon you O Trustee of Allah , I testify that you are His Messenger, and you are Muhammad son of Abdullah; and I testify that you had advised your community (Ummah) and that you have strived in the path of your Lord, and that you have worshipped Him until the end,so may Allah reward you, O Messenger of Allah, with the best reward a Prophet or a Messenger ever received on behalf of his Community, O Allah send blessings on Mohammad and the children of Mohammad more pleasant than the blessings on Ibrahim and the children of Ibrahim, verily You are worthy of praise and glory.”

It is narrated from Imam Ja’far As-Sadiq (as) that after every mandatory prayer he continued sitting at the place of the prayer till he cursed four accursed men andfour accursed women of the ummat.