$\int_1^{\infty}\left(\frac{x+2}{\sqrt{x^4-2}}\right)dx$

$\lim_{c\to\infty }\left(\frac{1}{2}\ln\left|\frac{c^{2}}{\sqrt{2}}+\frac{\sqrt{c^4-2}}{\sqrt{2}}\right|+\sqrt[4]{\left(2\right)^{3}}F\left(\frac{\arctan\left(\frac{\sqrt{c^4-2}}{\sqrt{2}}\right)}{2}\Big\vert 2\right)-\left(\frac{1}{2}\ln\left|\frac{1^{2}}{\sqrt{2}}+\frac{\sqrt{1^4-2}}{\sqrt{2}}\right|+\sqrt[4]{\left(2\right)^{3}}F\left(\frac{\arctan\left(\frac{\sqrt{1^4-2}}{\sqrt{2}}\right)}{2}\Big\vert 2\right)\right)\right)$