<s>[INST] How to evaluate these $4$ integrals? 
\begin{align}
(1) & \int \frac{\sqrt{x^4 + \frac{1}{x^4} + 2}}{x^3}dx\\
(2) & \int \sqrt{1 - \sin(2x)}dx & \forall x \in [-\pi/4, \pi/4]\\
(3) & \int \cos^2(2x) \cos(x)dx\\
(4) & \int \frac{dx}{\sqrt{x +1}(x+5)}
\end{align}

I'd be very grateful for any help and hints! 
 [/INST] $1$. $\sqrt{x^4 + \dfrac1{x^4} + 2} = x^2 + \dfrac1{x^2}$.
$2$. $\sqrt{1-\sin(2x)} = \sqrt{\cos^2(x) + \sin^2(x) -2 \sin(x) \cos(x)} = \cos(x) - \sin(x)\,\,\, \forall x \in [-\pi/4,\pi/4]$.
$3$. $\cos^2(2x) \cos(x) = \dfrac{1+\cos(4x)}2 \cdot \cos(x) = \dfrac{\cos(x)}2 + \dfrac{\cos(5x) - \cos(3x)}4$.
$4$. $\dfrac{dx}{\sqrt{x+1} (x+5)} \underset{\sqrt{x+1} = t}{\to} \dfrac{2tdt}{t(t^2+4)} = \dfrac{2dt}{t^2+4} = d \left(\arctan \left(\dfrac{t}2 \right)\right)$
</s>