In my research, I need to evaluate an integral: $$\int_{R^{3}}\frac{y^{3}}{(x\xi^{2}+y^{2})^{3}}\log(\xi^{2}+\frac{1}{4})d\xi$$ where $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it is a biharmonic function on the uphalf space or not. I also want to know will $$ \log(x^{2}+(y+\frac{1}{2})^{2})c\int_{R^{3}} \frac{y^{3}}{(x\xi^{2}+y^{2})^{3}} \log(\xi^{2}+\frac{1}{4})d\xi$$ be bounded on the up half space? Here $c$ is a constant such that $$c\int_{R^{3}}\frac{y^{3}}{(x\xi^{2}+y^{2})^{3}}d\xi=1$$
If you use spherical coordinates, the integral splits in a radial part and an integral over the unit sphere in $\mathbb{R}^3$. The spherical integral is $$ \int_{S^2} \frac{1}{y^2+r^2+x^2  2 r \langle x , \xi' \rangle} d \sigma(\xi') $$where $\xi= r \xi'$, $\xi' \in S^2$. Now you can apply the FunkHecke theorem to compute this spherical integral, because the integrand only depends on the inner product of $\xi'$ with a fixed vector. (you can find the FunkHecke theorem in books on spherical harmonics).  haven't carried out this computation explicitly, but this is how I would compute this integral. 

