This question is from the MIT Integration Bee 2023 Finals, and this is Question 3. This integral was to be solved within four minutes, and the goal is to show that $$\int_{-1/2}^{1/2} \sqrt{x^2+1+\sqrt{x^4+x^2+1}}\ \textrm{d}x = \frac{\sqrt{7}}{2\sqrt{2}} + \frac{3}{4\sqrt{2}} \log\left(\frac{\sqrt{7}+2}{\sqrt{3}}\right)$$
My first attempt was to rewrite the inside of the nested square root as $(x^2+1)^2 - x^2$, after which I performed the trigonometric substitution $x = \tan(\theta)$. That made the integral above transform to $$\int_{-\alpha}^{\alpha} \sqrt{\sec^2(\theta)+\sqrt{\sec^4(\theta)-\tan^2(\theta)}}\sec^2(\theta)\ \textrm{d}\theta$$
Here, $\alpha = \tan^{-1}(\frac{1}{2})$. From here, I attempted to force the inside of the nested square root into some $(a\pm b)^2$ form, but doing so required me to be in $\operatorname{GF}(2)$. In my assumption, that meant $\sec^4(\theta) - \tan^2(\theta) =(\sec^2(\theta) + \sec(\theta) + 1)^2$, and the integral would transform to $$\int_{-\alpha}^{\alpha} \sqrt{2\sec^2(\theta) + \sec(\theta) + 1}\sec^2(\theta)\ \textrm{d}\theta $$
I'm pretty sure I made a mistake somewhere, but I don't know where. In the event I haven't, how could I simplify the inside of the nested square root? Or, what are other methods on attacking this question? I don't know complex analysis.