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Integral of $$\int_{0}^{\frac{\pi}{2}}\tan^{-1}(\sin(x))$$ can be given in the following form (obtained from CAS):

$$ I = \text{Li}_2\left(-\frac{1}{\sqrt{2 }}\right)-\text{Li}_2\left(-1- \sqrt{2}\right)+\frac{\pi ^2}{12}+\frac{\log ^2(2)}{8} \\ -\log \left(1+\sqrt{2}\right) \log \left(2+\sqrt{2}\right) $$

It can be derived using integration under the differential sign (found here):

https://boredofstudies.org/threads/hsc-2018-2019-mx2-integration-marathon.368911/post-7318017

the approximate numerical result is: $$ I \approx 0.845291$$

I have also found that expression $$ \frac{\pi ^2}{8}-\frac{1}{2} \log ^2\left(1+\sqrt{2}\right) $$ yields the same numerical results to within thousands of digits.

Is it possible to find this closed-form solution without dilogarithm from this integral?

tpk
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  • Also see this one that links to the same post. – Lee David Chung Lin Feb 04 '20 at 19:19
  • Wow. Yes, I am bad at searching stackexchange. Sorry and thank you for help. – tpk Feb 04 '20 at 19:24
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    It's not that you're bad at it, it's that the search functionality on StackExchange certainly needs improvement. Meanwhile, in general, searching math on the internet is tricky. Many people, including me, have had good results using Approach0. It has some kinks but very powerful after one gets the hang of nailing down the correct input. – Lee David Chung Lin Feb 04 '20 at 19:30
  • Thank You very much. Finding out about Approacch0 was worth posting this, it looks great! Again many thanks. – tpk Feb 04 '20 at 19:53

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