Obeying A Corrupted Authority

 Let everyone be subject to the governing authorities, for there is no authority except that which God has established. The authorities that exist have been established by God.

                                                                                            ~ Romans 13:13

What if the governing authority is not exactly good? What if they are corrupt? What if they pass a law or rules that goes against God's commands? 
For example, the governing authority bans Christianity. Everyone are not allowed to worship Jesus, or go to such a place called church. Do you obey the authority? God says that authorities are established by God. We obey God by obeying the authority. Correct?
No. God commands us to follow the governing authority and obey their rules. But God is still the ultimate authority over us. Whatever we do, we answer eventually to God. 
I recall a story in 1 Samuel 8, where Israel demanded for a king to rule over them. 

4So all the elders of Israel gathered together and came to Samuel at Ramah. 5They said to him, “You are old, and your sons do not follow your ways; now appoint a king to lead8:5 Traditionally judge; also in verses 6 and 20 us, such as all the other nations have.”8:5 Traditionally judge; also in verses 6 and 20 us, such as all the other nations have.”8:5 Traditionally judge; also in verses 6 and 20 us, such as all the other nations have.”
6But when they said, “Give us a king to lead us,” this displeased Samuel; so he prayed to the Lord. 7And the Lord told him: “Listen to all that the people are saying to you; it is not you they have rejected, but they have rejected me as their king. 
Originally, God placed Himself to be our King. That sets me thinking; did God placed a human authority over us because humans wanted it? If humans obey God and did not reject God as our King, will God still place a human ruler over us?

What I do know is that God is our ultimate King, our Judge and Ruler. God's authority is above the earthly authority.

“Do you refuse to speak to me?” Pilate said. “Don’t you realize I have power either to free you or to crucify you?”
11Jesus answered, “You would have no power over me if it were not given to you from above.                                             ~ John 19:10-11

So, what should we, as a Christian do, when the governing authority asked us to do something immoral?

Firstly, determine whether the "rules" are indeed against God's commands. We should not follow if it is against God's commands. Sticking to our beliefs might put us in trouble, or in jail. Should we resist when we are caught? Surely not. 
We can refer to Daniel 3. Shadrach, Meshach and Abednego did not follow the king's command to worship the idol. So the king ordered for them to be thrown into the fiery furnace. They did not resist the punishment. They took the punishment. 

So to summarise :
We, as a Christian ultimately have to obey God's commands. 
If we disobey the governing authority in response to obeying God's commands and got caught for it, we do not run away from the punishment. 

I have this picture in my mind, as shown below :

Our actions to follow or not follow the rules set by the governing authority, determine our obedience to God. 

Our acceptance of the punishment, determine our obedience to the authority (which is an act of obedience to God).

I thank God for the people who helped me to answer this question and I thank God for clearing my doubts. I hope that it benefits those who desires to seek the truth.