<s>[INST] Clever way of calculating the integral $ \int \frac{dt}{t^2\sqrt{t-2} } $ 
$$ \int  \frac{\text{d}t}{t^2\sqrt{t-2} } $$

I know it can be calculated using somewhat complicated substitutions, but is there possibly some clever way of solving that type of integral? I don't even expect full solution, just ideas.
 [/INST] \begin{align}
u & = \sqrt{t-2} \\[8pt]
u^2 & = t-2 \\[8pt]
u^2 + 2 & = t \\[8pt]
2u\,du & = dt
\end{align}
\begin{align}
\int \frac{dt}{t^2\sqrt{t-2}} & = \int \frac{2u\,du}{(u^2+2)^2 u} = \int\frac{2\,du}{(u^2+2)^2}
\end{align}
Then let $u = \sqrt 2 \tan \theta$, $du= \sqrt 2 \sec^2\theta\,d\theta$, $u^2+2=2\sec^2\theta$.  The integral becomes
$$
\int \frac{2\sqrt 2 \sec^2\theta\,d\theta}{\sec^4\theta} = 2\sqrt2 \int\cos^2\theta\,d\theta, \quad\text{etc.} 
$$
</s>