I Was Vey Amaze At The Solution 3

Calculus Level 4

\[ \cfrac{i\left(\operatorname{Li}_2\left(\cfrac{i+1}{2}\right)-\operatorname{Li}_2\left(i\right)+\operatorname{Li}_2\left(-i\right)-\operatorname{Li}_2\left(-\cfrac{i-1}{2}\right)\right)}{2} \]

If the above can be expressed in the form of \(\cfrac{\pi \ln{(q)}}{r}\), where \(q\) and \(r\) are both positive integers and \(q\) being a prime, find \( q + r \).


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