/*Approximate solution approach, here a,b >=0, element of integers : 
/%For integrals from Aθpc=0 to final, at Aθpc=0, sin(Aθpc(POIo(t),t))=0 and the expression is 0, producing a zero lower result for definite integrals. 

This was an attempt to [check, improve] results by look at Rpcs(POIo(t),t) as a variable of integration "with respect to" (wrt) Aθpc.  However, that is not correct when following the circumference.

/* General result from (5) below : 
/%	∫[∂(Aθpc),0.to.Aθpcf: Rpcs(POIo(t),t)^(-β)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^a}  = 1/(a+β)/Rocs(POIo)*Rpcs(POIo(t),t)^(-β+1)*sin(Aθpc(POIo(t),t))^(a+1)

/* List of special integrals
/* 23Jun2016 not used in current versions as Rpcs(POIo(t)=0) is a "pseudo-constant" for integration wrt dAθpc
29Aug2019 - can switch from [∂(Aθpc),0.to.Aθpcf:, cos(Aθpc(POIo(t),t))]  to  [∂(Aθoc),0.to.Aθocf:, cos(Aθoc(POIp(t),t))]  etc...

/%	
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))							  *Rpcs(POIo(t),t)^(-4)]	= 1/ 4/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^1
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))							  *Rpcs(POIo(t),t)^(-5)]	= 1/ 5/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^1
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-3)]	= 1/ 4/Rocs(POIo)*Rpcs(POIo(t),t)^(-2)*sin(Aθpc(POIo(t),t))^2
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-5)]	= 1/ 6/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^2
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-4)]	= 1/ 6/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^3
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)]	= 1/ 7/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^3
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-4)]	= 1/ 7/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^4
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-6)]	= 1/ 9/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^4
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-7)]	= 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-5)]	= 1/ 9/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^5
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)]	= 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^5
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-5)]	= 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^6
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-6)]	= 1/11/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^6
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-7)]	= 1/12/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^6
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-6)]	= 1/12/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^7
(endMath)
(mathH)  ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^7*Rpcs(POIo(t),t)^(-7)]	= 1/14/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^8
(endMath)


/*--   
First attempt :  
/%	    ∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(a + 1)*Rpcs(POIo(t),t)^( - β + 1)] 
	=   ∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(a + 1)]                                   *Rpcs(POIo(t),t)^(-β+1)  
	  +            sin(Aθpc(POIo(t),t))^(a+1)                          *∂[∂(Aθpc): Rpcs(POIo(t),t)^( - β + 1)]
(1)	=   (a+1)*sin(Aθpc(POIo(t),t))^(a  )*∂[∂(Aθpc): sin(Aθpc(POIo(t),t))]            *Rpcs(POIo(t),t)^(-β+1)  
	  +            sin(Aθpc(POIo(t),t))^(a+1) *(-β+1)*Rpcs(POIo(t),t)^(-β)*∂[∂(Aθpc): Rpcs(POIo(t),t)]


From "∂[∂(Aθpc): Rpcs(POIo(t),t)] = ∂[∂(Aθpc): |Rpcv(POIo(t),t)|]"
2*) 	∂[∂(Aθpc): Rpcs(POIo(t),t)]  =  (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t))

Subbing (2*) into (1) : 
	=   ( a+1)*sin(Aθpc(POIo(t),t))^(a  )*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β+1)  
	  + (-β+1)*sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β)*(-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t))
	=   ( a+1)                *Rpcs(POIo(t),t)^(-β+1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a  )      
	  + (-β+1)*(-1)*Rocs(POIo)*sin(Aθoc(POIo))*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a+1+1-2)
	=   ( a+1)                *Rpcs(POIo(t),t)^(-β  )                     *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a  )    *Rpcs(POIo(t),t)^(-β+1)  
	  + ( β-1)*Rocs(POIo)*sin(Aθoc(POIo))     *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1)    *Rpcs(POIo(t),t)^(-β  )
(2)	=   cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)*Rpcs(POIo(t),t)^(-β)  
	    * [ ( a+1)*Rpcs(POIo(t),t) + ( β-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-1) }

From "Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants" : but thisd is with respect to time!!
With respect to Aθpc(POIo(t),t), these are NOT constants (i.e POIo changes along trajectory of particle?)
5*) 	  Rpcs (POIo(t),t)*sin(Aθpc(POIo(t),t))  = Rocs(POIo)*sin(Aθoc(POIo))
At t=0, the observer & particle reference frames coincide, and 
3a)	sin(Aθoc(POIo))|t=0  =  sin(Aθpc(POIo(t),t))^(-1)|t=0
3b)	    Rocs(POIo) |t=0  =  Rpcs(POIo(t),t)    

Subbing (5*),(3a),(3b) into (2) : 
(4)	                 ∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(a + 1)*Rpcs(POIo(t),t)^( - β + 1)] 
	=         cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)  *Rpcs(POIo(t),t)^(-β) *[ (a+1)*Rpcs(POIo(t),t) + (β-1)*Rocs(POIo) }
	=         cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)  *Rpcs(POIo(t),t)^(-β) *[ (a+1) + (β-1) ]*Rpcs(POIo(t),t)
	=         cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)  *Rpcs(POIo(t),t)^(-β) *  (a+β)*Rpcs(POIo(t),t)
	=   (a+β)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)  *Rpcs(POIo(t),t)^(-β+1)
For t=0, 
a,β >=0, ∈ integers.  Also, at Aθpc=0, the expression is 0. 
As the upper endpoint of the integration at time t=0, Aθpc(POIo(t),t)), Rocs(POIo)

Therefore : 
(5)	∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)*Rpcs(POIo(t),t)^( - β)}  =   1/(a+β)*sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β+1)
For t=0 when RFp & RFo coincide, a,β >=0, integers.  POIo does NOT have to be on the perpendicular running through the origins.
For integrals from Aθpc=0 to final, at Aθpc=0, sin(Aθpc(POIo(t),t))=0 and the expression is 0, producing a zero lower result for definite integrals. 
At Aθpc(POIo(t),t)) = 0, the expression is zero so the lower limit of integraion drops out. 

HOWEVER, at the upper limit of integration, @Aθpc(POIo(t),t)), Rpcs(POIo(t),t) = Rocs(POIo), so we cancel out their powers, result as Rpcs(POIo(t),t)

(6)	∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)*Rpcs(POIo(t),t)^( - β)}  =   1/(a+β)*sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β)
For t=0 when RFp & RFo coincide, a,β >=0, integers.  POIo does NOT have to be on the perpendicular running through the origins.
For integrals from Aθpc=0 to final, at Aθpc=0, sin(Aθpc(POIo(t),t))=0 and the expression is 0, producing a zero lower result for definite integrals. 
At Aθpc(POIo(t),t)) = 0, the expression is zero so the lower limit of integraion drops out. 


/*-----+
Check on : 
/%∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^( - 7)}  
	=   sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β+1)/(a+β)/Rocs(POIo)
where a=3, β=7
	=   sin(Aθpc(POIo(t),t))^(3+1)*Rpcs(POIo(t),t)^(-7+1)/(a+β)/Rocs(POIo)
(1)	=   sin(Aθpc(POIo(t),t))^(4)  *Rpcs(POIo(t),t)^(-6)  /(10 )/Rocs(POIo)

Taking the derivative wrt Aθpc of (1) : 
(2)	   ∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^( - 6)/10/Rocs(POIo)] 
	=  [ 1/10/Rocs(POIo) }*∂[∂(Aθpc): sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^( - 6)] 

/*--+ Looking at the derivative 
	   dp[dAθpc : sin(Aθpc(POIo(t),t))^(4)   *Rpcs(POIo(t),t)^(-6) ] 
	=  dp[dAθpc : sin(Aθpc(POIo(t),t))^(4) ] *Rpcs(POIo(t),t)^(-6)   + sin(Aθpc(POIo(t),t))^(4)*dp[dAθpc : Rpcs(POIo(t),t)^(-6) ] 
	=   4   *sin(Aθpc(POIo(t),t))^(3)*dp[dAθpc : sin(Aθpc(POIo(t),t))  ]  *Rpcs(POIo(t),t)^(-6)   
	  +      sin(Aθpc(POIo(t),t))^(4)*(-6)*Rpcs(POIo(t),t)^(-7)*dp[dAθpc : Rpcs(POIo(t),t) ]
(3)	=   4   *sin(Aθpc(POIo(t),t))^(3)*cos(Aθpc(POIo(t),t))                *Rpcs(POIo(t),t)^(-6)   
	  + (-6)*sin(Aθpc(POIo(t),t))^(4)                                  *Rpcs(POIo(t),t)^(-7)*dp[dAθpc : Rpcs(POIo(t),t) ]

/* From "dp[dAθpc : Rpcs(POIo(t),t) ] = dp[dAθpc : |Rpcv(POIo(t),t)|]"
2*) 	dp[dAθpc : Rpcs(POIo(t),t)]  =  (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t))

Subbing into (3) : 
	=   4   *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-6)   
	  + (-6)                  *sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-7)*(-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t))
	=   4   *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-6)   
	  + 6   *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(2)*Rpcs(POIo(t),t)^(-7)     *Rocs(POIo)*sin(Aθoc(POIo))
(4)	=        cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7)   
	    *[ 4*Rpcs(POIo(t),t)  + 6*sin(Aθpc(POIo(t),t))^(-1)                      *Rocs(POIo)*sin(Aθoc(POIo)) }  

As in the previous sub-sub-section : 
From "Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants" : 
5*) 	  Rpcs (POIo(t),t)*sin(Aθpc(POIo(t),t))  = Rocs(POIo)*sin(Aθoc(POIo))
At t=0, the observer & particle reference frames coincide, and 
3a)	sin(Aθoc(POIo))|t=0  =  sin(Aθpc(POIo(t),t))^(-1)|t=0
3b)	    Rocs(POIo) |t=0  =  Rpcs(POIo(t),t)    

Subbing into (4) : 
	          dp[dAθpc : sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-6) ] 
	=  cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7)*Rocs(POIo)*( 4 + 6 )  
	=  cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7)*10*Rocs(POIo)

Integrating both sides :  
	∫[dAθpc, 0 to Aθpcf : cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7)*10*Rocs(POIo) }
	=  sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-6)
Removing constants from within the integral : 
	∫[dAθpc, 0 to Aθpcf : cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7) }  = sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-6)/10/Rocs(POIo)

This is the same as the starting point at the start of this example :
	∫[dAθpc, 0 to Aθpcf : cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(3)*Rpcs(POIo(t),t)^(-7) }  = sin(Aθpc(POIo(t),t))^(4)*Rpcs(POIo(t),t)^(-6)/10/Rocs(POIo)
confiming the general result for this example.


/*-----+
Try this on expression missing a term : 
This example VIOLATES a condition for the general solution in that the sin(Aθpc(POIo(t),t))^(a) term is missing (a=0, when it should be >0 integer).
However, I'm curious to try it out anyways.
/*
/%	∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a)*Rpcs(POIo(t),t)^( - β)}
	=   sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β+1)/(a+β)/Rocs(POIo)
/*For t=0 when RFp & RFo coincide, a,b >=0, integers.  POIo does NOT have to be on the perpendicular running through the origins.
For integrals from Aθpc=0 to final, at Aθpc=0, sin(Aθpc(POIo(t),t))=0 and the expression is 0, producing a zero lower result for definite integrals. 
/%At Aθpc(POIo(t),t)) = 0 & π/2, the expression is zero. 

/* For this example :
/%	∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^( - 5)}
	=   sin(Aθpc(POIo(t),t))^(a+1)*Rpcs(POIo(t),t)^(-β+1)/(a+β)/Rocs(POIo)
where a=0, β=5
(1)	=   sin(Aθpc(POIo(t),t))^(1)*Rpcs(POIo(t),t)^(-4)/5/Rocs(POIo)

/*--+ Taking the derivative of the solution : 
	    dp[dAθpc : sin(Aθpc(POIo(t),t))^(1)                              *Rpcs(POIo(t),t)^(-4)/5/Rocs(POIo) ] 
	=   1/5/Rocs(POIo)*
	 {  dp[dAθpc : sin(Aθpc(POIo(t),t))^(1)]                             *Rpcs(POIo(t),t)^(-4) ] 
	  +            sin(Aθpc(POIo(t),t))^(1)                   *dp[dAθpc : Rpcs(POIo(t),t)^(-4) ]
	 }
(2)	=   1/5/Rocs(POIo)*
	 {  (1) *sin(Aθpc(POIo(t),t))^(0)*dp[dAθpc : sin(Aθpc(POIo(t),t))]      *Rpcs(POIo(t),t)^(-4) 
	  +      sin(Aθpc(POIo(t),t))^(1)                               *(-4)*Rpcs(POIo(t),t)^(-5)*dp[dAθpc : Rpcs(POIo(t),t) ] 
	 }
/* From "dp[dAθpc : Rpcs(POIo(t),t) ] = dp[dAθpc : |Rpcv(POIo(t),t)|]"
2*) 	dp[dAθpc : Rpcs(POIo(t),t)]  =  (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t))

Subbing (2*) into (2) :  
	=   1/5/Rocs(POIo)*
	 {       sin(Aθpc(POIo(t),t))^(0)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) 
	  +      sin(Aθpc(POIo(t),t))^(1)*(-4)*Rpcs(POIo(t),t)^(-5)*(-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t)) 
	 }
	=   1/5/Rocs(POIo)*
	 {            cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4)  
	  + (-4)*(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(-1)*Rpcs(POIo(t),t)^(-5)*Rocs(POIo)*sin(Aθoc(POIo))
	 }
Factor out 
	=   1/5 /Rocs(POIo)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)*
	 {       Rpcs(POIo(t),t)  
	  + 4                                                    *sin(Aθpc(POIo(t),t))^(-1)*Rocs(POIo)*sin(Aθoc(POIo))
	 }
	=   1/5 /Rocs(POIo)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)*
	 {       Rpcs(POIo(t),t)                                +  4*sin(Aθpc(POIo(t),t))^(-1)*Rocs(POIo)*sin(Aθoc(POIo))  }
From "Rocv(POIo), Aθoc(POIo), Aφoc(POIo) are constants" : 
5*) 	  Rpcs (POIo(t),t)*sin(Aθpc(POIo(t),t))  = Rocs(POIo)*sin(Aθoc(POIo))
At t=0, the observer & particle reference frames coincide, and 
3a)	sin(Aθoc(POIo))|t=0  =  sin(Aθpc(POIo(t),t))^(-1)|t=0
3b)	    Rocs(POIo) |t=0  =  Rpcs(POIo(t),t)    
	=   1/5/Rocs(POIo)                *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)*{  Rpcs(POIo(t),t)  +  4*Rocs(POIo) }
	=   1/5/Rocs(POIo)                *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)*Rocs(POIo)*{ 1 + 4 }
	= { 1/5/Rocs(POIo)*Rocs(POIo)*5 } *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)
	=                                  cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)

Summarizing : 
(3)	    dp[dAθpc : sin(Aθpc(POIo(t),t))^(1)*Rpcs(POIo(t),t)^(-4)/5/Rocs(POIo) ]
	=   cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5)

/*Which is the same as the integral term of (1), so the solution is correct.