# Reducing a long complex expression to a simplier one by using assumptions

so I have an ugly looking equation that I wish to solve under some specific circumstances. Using DSolve I write;

 t = DSolve[{p'[r] ==  A*((r - (rc^2)/r) - ((l)*Log[((rc - r)^2 + l)/l])/r +
(2*Sqrt[l]*rc*ArcTan[(rc - r)/Sqrt[l]])/r), p[rc] == 0},  p[r], r]


And this yields a long, ugly expression involving imaginary numbers and PolyLog functions - this is fair enough, but I'm trying to simplify it further by telling it only to look at realistic cases; $A, r, r_{c}$ are always real and positive, as is $p$. Also, $r_{c} >= r$ so that $r_{c} - r$ is always a positive number. $L$ is also always positive and orders of magnitude smaller than $r$ or $r_{c}$. So I attempted to use full simplify with these assumptions on the expression by

u = FullSimplify[t, rc > 0 && r > 0 && rc >= r && l > 0 && A > 0 && p >= 0 ]


This simplifies the expression somewhat, but still has clunky imaginary parts and I suspect it can be further simplified. I have tried Refine but this doesn't yield a better expression. Is it possible to clean this up, or do I accept this as just a complex looking expression that gives me real answers and cannot be further simplified ? Thanks!

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