<s>[INST] Evaluation of $\int_0^1 \frac{x^3}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,\mathrm dx$ How does one evaluate the following integral?
$$\int_0^1 \frac{x^3}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,\mathrm dx$$
This is a homework problem and I have been evaluating this integral for hours yet no success so far. I have tried to rationalize the integrand by multiplying it with
$$\frac{2(2-x^2)(1+x^2) - 3\sqrt{(2-x^2)(1+x^2)}}{2(2-x^2)(1+x^2) - 3\sqrt{(2-x^2)(1+x^2)}}$$
but the integrand is getting worse. I have tried to use trigonometric substitutions like $x=\tan\theta$ and $x=\sqrt{2}\sin\theta$, but I cannot rid off the square root form. I have also tried to use hyperbolic trigonometric substitutions but the thing does not get any easier neither also substitution $y=x^2$ nor $y=\sqrt{(2-x^2)(1+x^2)}$. Using integration by parts is almost impossible for this one. I have also tried to use the tricks from this thread, but still did not get anything. No clue is given. My professor said, we must use clever substitutions but I cannot find them. Any idea or hint? Any help would be appreciated. Thanks in advance.
Edit :
The answer I got from my Prof is $\dfrac{3-2\sqrt{2}}{6}$.
 [/INST] If you consider these imaginary parts of elliptic integrals as closed-forms, then
$$\frac{1}{2}-\frac{\sqrt 2}{3}-\frac{\sqrt{2} \log
   \left(\sqrt{2}-1\right)}{8}+\frac{\sqrt{2} \log
   \left(\sqrt{2}+1\right)}{8}-\frac{\sqrt 2}{8}\Im\left(F\left(i \log\left(\sqrt 2+1\right)|-\tfrac{1}{2}\right)\right)+\frac{\sqrt{2}}{6}\Im\left(E\left(i \log\left(\sqrt 2+1\right)|-\tfrac{1}{2}\right)\right)-\frac{3\sqrt{2}}{8
   }\Im\left(\Pi
   \left(-2;i \log\left(\sqrt 2+1\right)|-\tfrac{1}{2}\right)\right),$$
where $F\left(x\,|\,m\right)$ is the elliptic integral of the first kind, $E\left(x\,|\,m\right)$ is the elliptic integral of the second kind, and $\Pi\left(n; x \, | \, m\right)$ is the elliptic integral of the third kind with parameters $m=k^2$.
</s>