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I find the integral

Integrate[Sqrt[2 x - x^2] + x^3 Log[(9 - x^2)/(9 + x^2)], {x, 0, 2}]

and got

1/2 (-36 + π + 65 ArcTanh[4/9])

How can I write this answer has the form $\dfrac{65}{4} \ln \dfrac{13}{5} + \dfrac{\pi}{2}-18$

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expr = 1/2 (-36 + \[Pi] + 65 ArcTanh[4/9]) // TrigToExp

(*   -18 + \[Pi]/2 + 65/4 Log[13/9] + 65/4 Log[9/5]   *)


MapAt[HoldForm, expr, {{1}, {2}}] // Simplify // ReleaseHold

(*   -18 + \[Pi]/2 + 65/4 Log[13/5]   *)
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A little more general approach:

expr = 1/2 (-36 + \[Pi] + 65 ArcTanh[4/9])

expr /. x : ArcTanh[_] :> Simplify[TrigToExp[x]]
1/2 (-36 + \[Pi] + 65/2 Log[13/5])

About simplifying expressions with Log take a look here too.

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