Why Wolfram Cloud doesn't simplify below BBP type sum, which I came up with?

Question

Why Wolfram Cloud don't simplify below BBP type sum, which I came up with:

simplify Sum((1/81)^(k+1)*(9/(2*k+1)+1/(2*k+2)),k = 0 .. infinity)

to

2*log(3) - 3*log(2)

?

I would expect to see in the output

2*log(3) - 3*log(2) 

or at least

log(9)/log(8)

but instead the output is:

ArcTanh[1/9]+1/2 Log[81/80] 

?

PS Here are the messages I have received on WOLFRAM COMMUNITY where I also posted my question:

1. With Mathematica I get simplification with FullSimplify, but not with Simplify alone.

2. Both Wolfram Alpha and Wolfram Cloud are powerful computational platforms, but there could be several reasons why a specific mathematical expression might not simplify as expected:

Algorithmic Limitations: While these platforms have advanced algorithms, there are cases where certain simplifications aren't recognized, especially for non-standard or less common forms.

Input Interpretation: The way you've input the expression might lead the software to interpret or process it differently than intended.

Complexity: Some expressions, even if they have simpler forms, might be perceived as computationally expensive to evaluate or simplify, leading to timeouts or incomplete.

Answer 1

When I try

simplify Sum((1/81)^(k+1)(9/(2k+1)+1/(2*k+2)),k = 0 .. infinity)

in WolframAlpha

it shows me three different forms of the result, the first one is log(9/8)

When I try Wolfram Mathematica in the cloud

FullSimplify[Sum[(1/81)^(k+1)(9/(2k+1)+1/(2*k+2)),{k,0,Infinity}]]

it returns

Log[9/8]

Are you and I doing exactly the same thing and getting different results? Or is this just the form of the output made you think we were getting different results?