Evaluate the integral

${\displaystyle \underset{\sqrt[\phantom{2}]{2}}{\overset{2}{}}}\frac{1}{t^3\sqrt[\phantom{2}]{t^2-1}}dt$

Select the correct answer.

$\frac{}{12}+\frac{\sqrt[\phantom{2}]{3}}{8}-\frac{1}{4}$

$\frac{}{24}+\frac{\sqrt[\phantom{2}]{3}}{4}-\frac{1}{4}$

$\frac{}{24}+\frac{\sqrt[\phantom{2}]{3}}{8}-\frac{1}{4}$

$\frac{}{12}+\frac{\sqrt[\phantom{2}]{3}}{4}-\frac{1}{4}$