$\int_0^1 \sqrt {\tan^{-1}x}\space dx=$? How to evaluate $$\int_0^1 \sqrt {\tan^{-1}x}\space dx\qquad ?$$ Is there an elementary expression or value for it? (though I know that there is no elementary expression for $\int \sqrt {\tan^{-1}x}\space dx$)  .
You can have a solution as an infinite sum


  $$\int_0^1 \sqrt {\tan^{-1}x}\space dx = \frac{\sqrt {\pi }}{2}-\frac{1}{\sqrt {\pi }}\sum _{n=0}^{\infty }{\frac {\psi^{(2\,n+1
)} \left( \frac{1}{2} \right) }{ 16^n\left( 4\,n+3 \right)  \left( 2\,
n+1 \right) !}} \sim 0.6298233443,$$


where $\psi(x)$ is the digamma function.
By using the formula http://upload.wikimedia.org/wikipedia/en/math/7/2/a/72a1058ad2087aec467af24bddcf9479.png, we have $\int_0^1\sqrt{\tan^{-1}x}~dx=\sum\limits_{n=1}^\infty\sum\limits_{m=1}^{2^n-1}\dfrac{(-1)^{m+1}}{2^n}\sqrt{\tan^{-1}\dfrac{m}{2^n}}$