## Appendix A
 \tag{12}\]  
where  
\[V_{1}=-\frac{Ap_{1}+B}{p_{2}-p_{1}}\,\ \ V_{2}=-\frac{Ap_{2}+B}{p_{2}-p_{1}}\]The simplified form is  
\[ \begin{split} I_{2}^{(3)}=-\frac{\eta^{2}R}{2}J_{10}-\frac{k_{1}\eta \sqrt{\eta^{2}+1}\sqrt{RR_{1}}}{4}\int\limits_{0}^{\frac{\pi}{2}}\Bigl{\{} \cos^{2}(2\beta)+a_{22}\cos(2\beta)-1+a_{33}\Bigr{\}}\frac{d\beta}{\Delta_{1}}- \\ \qquad\qquad\frac{k_{1}\eta\sqrt{\eta^{2}+1}\sqrt{RR_{1}}}{4} \int\limits_{0}^{\frac{\pi}{2}}\frac{(Acos(2\beta)+B)d\beta}{(\cos(2\beta)-p_ {1})(\cos(2\beta)-p_{2})\Delta}=-\frac{\eta^{2}R}{2}J_{10}-\\ \qquad\frac{k_{1}\eta\sqrt{\eta^{2}+1}\sqrt{RR_{1}}}{4}\int \limits_{0}^{\frac{\pi}{2}}\Bigl{\{}4\sin^{4}(\beta)-(2a_{2}+4)\sin^{2}(\beta) +a_{2}+a_{3}\Bigr{\}}\frac{d\beta}{\Delta_{1}}-\\ \qquad\frac{k_{1}\eta\sqrt{\eta^{2}+1}\sqrt{RR_{1}}}{4}\Biggl{\{} V_{1}\int\limits_{0}^{\frac{\pi}{2}}\frac{d\beta}{(\cos(2\beta)-p_{1})\Delta_{1}}+V_{2} \int\limits_{0}^{\frac{\pi}{2}}\frac{d\beta}{(\cos(2\beta)-p_{2})\Delta_{1}} \Biggr{\}}=-\frac{\eta^{2}R}{2}J_{10}-\\ \qquad\frac{k_{1}\eta\sqrt{\eta^{2}+1}\sqrt{RR_{1}}}{4}\Biggl{\{} 4\Bigl{[}\frac{2+k_{1}^{2}}{3k_{1}^{4}}K(k_{1})-\frac{2+2k_{1}^{2}}{3k_{1}^{4} }E(k_{1})\Bigr{]}-\frac{(2a_{22}+4)}{k_{1}^{2}}[K(k_{1})-E(k_{1})]-\\ \qquad(a_{22}+a_{33})\,K(k_{1})\bigr{\}}-\frac{k_{1}\eta\sqrt{\eta ^{2}+1}\sqrt{RR_{1}}}{4}\Bigl{[}\frac{V_{1}}{1-p_{1}}\Pi(h_{1},k_{1})+\frac{V _{2}}{1-p_{2}}\Pi(h_{2},k_{1})\Bigr{]}=\\ \qquad=-\frac{\eta^{2}R}{2}J_{20}-\frac{\eta\sqrt{\eta^{2}+1}\sqrt {RR_{1}}}{2k_{1}^{4}}\bigl{\{}\bigl{[}3(a_{22}+a_{33})k_{1}^{4}-(6a_{22}+8)k_ {1}^{2}+8\bigr{]}\,K(k_{1})+\\ \bigl{[}(6a_{22}+4)k_{1}^{2}-8\bigr{]}\,E(k_{1})\bigr{\}}+\frac{k_ {1}\eta\sqrt{\eta^{2}+1}\sqrt{RR_{1}}}{4}\Bigl{[}\frac{V_{1}}{1-p_{1}}\Pi(h_{ 1},k_{1})+\frac{V_{2}}{1-p_{2}}\Pi(h_{2},k_{1})\Bigr{]}\end{split}  \tag{10}\]  
with  
\[h_{1}=\frac{2}{1-p_{1}}\,,\,\,\,h_{2}=\frac{2}{1-p_{2}}\]  
Similarly,  
\[ \begin{split} I_{2}^{(4)}=\int\limits_{0}^{\frac{\pi}{2}}\bigl{[} \eta^{2}R\cos(2\beta)-R_{1}+\eta\bigl{(}z_{Q}-z_{1}\bigr{)}\bigr{]}\cos(2\beta )\text{arsinh}\frac{\eta R\cos(2\beta)+\eta R_{1}+z_{Q}-z_{2}}{\sqrt{D_{10}}} d\beta+\\ \frac{\eta^{2}R}{2}J_{20}\end{split} \]  
can be obtained in the following form. \[ \begin{split} I_{2}^{(4)}=\frac{\eta^{2}R}{2}J_{20}+\frac{\eta \sqrt{\eta^{2}+1}\sqrt{RF_{1}}}{12k_{2}^{3}}\bigl{\{}\bigl{[}3(a_{222}+a_{333}) k_{2}^{4}-(6a_{222}+8)k_{2}^{2}+8\bigr{]}\,K(k_{2})+\\ \bigl{[}(6a_{222}+4)k_{2}^{2}-8\bigr{]}\,E(k_{2})\bigr{\}}+\frac{k _{2}\eta\sqrt{\eta^{2}+1}\sqrt{RF_{1}}}{4}\Bigl{[}\frac{V_{3}}{1-p_{3}}\Pi(h_ {3},k_{2})+\frac{V_{4}}{1-p_{4}}\Pi(h_{4},k_{2})\Bigr{]}\end{split}  \tag{11}\]  
where  
\[ \begin{split} k_{2}^{2}=\frac{4 Rr_{1}}{\left(R+r_{1}\right)^{2}+ \bigl{(}z_{Q}-z_{2}\bigr{)}^{2}}\\ a_{111}=\frac{2\bigl{[}r_{1}-\eta\bigl{(}z_{Q}-2\bigr{)}\bigr{]} }{\eta^{2}R}\\ a_{222}=\frac{\eta R^{2}+\eta\bigl{(}z_{Q}-z_{2}\bigr{)}^{2}-r_{1} \bigl{(}z_{Q}-z_{2}\bigr{)}}{\eta Rr_{1}}\\ a_{333}=\frac{\bigl{[}r_{1}-\eta\bigl{(}z_{Q}-z_{2}\bigr{)} \bigr{]}^{2}+\bigl{(}\eta^{2}+1\bigr{)}R^{2}}{\eta^{2}R^{2}}\\ \frac{Cos(2\beta)+D}{\left[cos^{2}(2\beta)-a_{11}cos(2\beta)-a_ {33}\right]}=\frac{Cos(2\beta)+D}{\left(\cos(2\beta)-p_{3}\right)(cos(2\beta) -p_{4})}=\\ \frac{V_{3}}{cos(2\beta)-p_{3}}+\frac{V_{4}}{cos(2\beta)-p_{4}} \end{split} = \tag{12}\]  
\[ \begin{split} C=\left(a_{222}+a_{111}\right)a_{333}-a_{223}\,\,\,, \qquad D=a_{333}^{2}-a_{333}+a_{111}a_{222},\end{split}