<s>[INST] Evaluation of a particular type of integral involving logs and trigonometric function Is there any closed form for 
$$ \int _0 ^{\infty}\int _0 ^{\infty}\int _0 ^{\infty} \log(x)\log(y)\log(z)\cos(x^2+y^2+z^2)dzdydx$$
if yes then how to prove it?
 [/INST] One way to attack this is to exploit the symmetry of the integral.  Start by expanding the cosine term into individual pieces, i.e.,
$$\begin{align}\cos{(x^2+y^2+z^2)} &= \cos{x^2} \cos{(y^2+z^2)} - \sin{x^2} \sin{(y^2+z^2)}\\ &= \cos{x^2} \cos{y^2} \cos{z^2} - \cos{x^2} \sin{y^2} \sin{z^2}  \\ &\quad -\sin{x^2} \sin{y^2} \cos{z^2} - \sin{x^2} \cos{y^2} \sin{z^2} \end{align}$$
Next define
$$C = \int_0^{\infty} dx \, \cos{x^2} \: \log{x} $$
$$S = \int_0^{\infty} dx \, \sin{x^2} \: \log{x} $$
Then the triple integral is equal to $C^3-3 C S^2$. 
We obtain $C$ and $S$ using Cauchy's theorem to find $C+i S$.  We do this by considering the integral
$$\oint_{\eta} dz \, e^{i z^2} \log{z}$$
where $\eta$ is a circular wedge of radius $R$ and angle $\pi/4$ in the upper half plane with base along the positive real axis.  One may show that the integral over the circular arc vanishes as $\pi \log{R}/(4 R)$ as $R \to \infty$.  Thus, we have
$$C+i S = e^{i \pi/4} \int_0^{\infty} dt \, e^{-t^2} \left (\log{t} + i \frac{\pi}{4} \right ) $$
One may derive the value of this integral by noting that
$$\int_0^{\infty} dt \, e^{-t^2} \log{t} = -\frac{\sqrt{\pi}}{4} (\gamma + 2 \log{2})$$
This value is derived from using the value of 
$$\frac12 \left [\frac{d}{d\alpha} \Gamma \left (\frac{\alpha+1}{2} \right ) \right ]_{\alpha=0} = \frac14 \Gamma \left (\frac12 \right ) \psi \left ( \frac12 \right )$$
We also use
$$\int_0^{\infty} dt \, e^{-t^2} = \frac{\sqrt{\pi}}{2}$$
to obtain
$$C = -\frac18 \sqrt{\frac{\pi}{2}} (\pi + 2 \gamma + 4 \log{2})$$
$$S = \frac18 \sqrt{\frac{\pi}{2}} (\pi - 2 \gamma - 4 \log{2})$$
and therefore the triple integral is
$$C^3-3 C S^2 = \frac{\pi ^{3/2} \left(4 \gamma ^2-8 \gamma  \pi +\pi ^2+16 \log{2} \,(\gamma
   -\pi +\log{2})\right) (2 \gamma +\pi +4 \log{2})}{512 \sqrt{2}}$$
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