<s>[INST] Some Log integrals related to Gamma value Two years ago I evaluated some integrals related to $\Gamma(1/4)$.
First example:
$$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{2}\pi^{3/2}}{2\Gamma{(1/4)^{2}}}.$$
The proof I have is based on the following formula concerning the elliptic integral of first kind (integrating both sides with carefully).
$$i \cdot K(\sqrt{\frac{2k}{1+k}})=K(\sqrt{\frac{1-k}{1+k}})-K(\sqrt{\frac{1+k}{1-k}})\cdot\sqrt{\frac{1+k}{1-k}}$$
for $0<k<1$.
\begin{align}
(2)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{2x-1}-2x  \arctan{(\sqrt{2x-1})}}{\sqrt{x(1-x)}(2x-1)^{3/2}}dx=\frac{\sqrt{2}\pi^{5/2}}{\Gamma{(1/4)}^2}-\frac{\sqrt{2\pi}\Gamma{(1/4)}^2}{8}.
\end{align}
\begin{align}
(3)\hspace{.2cm}\int_{0}^{\pi/2}\frac{\sin{x}\log{(\tan{(x/2))}+x}}{\sqrt{\sin{x}}(\sin{x}+1)}dx=\pi-\frac{\sqrt{2\pi}\Gamma{(1/4)}^{2}}{16}-\frac{\sqrt{2}\pi^{5/2}}{2\Gamma{(1/4)}^{2}}.
\end{align}
Could you find a solution to (2) and (3) employing only Beta function or other method?
I've tried with Mathematica, Mapple, etc and seems that this evaluations are not so well known.
Question is an improvement of (1) that has been proved in an elementary approach.
 [/INST] I was hoping to finish it but I am stuuck with a last integral. Then, this is just a comment.
Consider
$$I(a)=\int_{0}^{1}\frac{\sqrt{x}\log{(a+\sqrt{1+x})}}{\sqrt{1-x^2}}\, dx$$
$$I'(a)=\int_{0}^{1} \frac{\sqrt{x}}{\sqrt{1-x^2} \left(a+\sqrt{x+1}\right)}\,dx$$
$$I'(a)=\pi-\pi  \sqrt{1+\frac{1}{a^2-2}}-2 a K(-1)+2 a\, \Pi
   \left(\left.\frac{1}{a^2-1}\right|-1\right)$$
$$I(1)=\pi-K(-1) -\sqrt{\frac{\pi }{2}} \Gamma \left(\frac{3}{4}\right)^2+2\color{red}{\int_0^1  a\, \Pi
   \left(\left.\frac{1}{a^2-1}\right|-1\right)\,da}$$
$$I(0)=\frac 12\int_0^1 \frac{\sqrt{x} \log (x+1)}{\sqrt{1-x^2}}\,dx$$
$$I(0)=-\sqrt{\pi }
   (\gamma +\log (2))\frac{ \Gamma \left(\frac{3}{4}\right)}{\Gamma
   \left(\frac{1}{4}\right)}+$$ $$\frac{\pi ^{3/2}
  }{4 \sqrt{2}} \text{HypergeometricPFQRegularized}^{(\{0,0,0\},\{0,1\},0)}\left(
   \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\},\left\{2,\frac
   {1}{2}\right\},\frac{1}{2}\right)$$
</s>