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I found the following formula in "INTEGRALS AND SERIES, vol.3" by Prudnikov, Brychkov and Marichev (page 188, eq.5). $$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{...

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{...