1
$\begingroup$

Evaluate $\int \dfrac{1}{\left(1+x^4\right)\left(\sqrt{\sqrt{1+x^4}}-x^2\right)}dx$

My attempt is as follows:-

$$x^2=\tan\theta$$ $$2xdx=\sec^2\theta d\theta$$ $$dx=\dfrac{\sec^2\theta d\theta}{2\sqrt{\tan\theta}}$$

$$\int \dfrac{1}{\left(1+\tan^2\theta\right)\left(\sqrt{\sec\theta}-\tan\theta\right)}\cdot\dfrac{\sec^2\theta}{2\sqrt{\tan\theta}}d\theta$$

$$\dfrac{1}{2}\cdot\int\dfrac{d\theta}{\sqrt{\tan\theta}\left(\sqrt{\sec\theta}-\tan\theta\right)}$$ $$\dfrac{1}{2}\cdot\int\dfrac{(\cos\theta)^\frac{3}{2} d\theta}{\sqrt{\sin\theta}\left(\sqrt{\cos\theta}-\sin\theta\right)}$$

$$\sqrt{\sin\theta}=y$$ $$\dfrac{\cos\theta}{2\sqrt{\sin\theta}}d\theta=dy$$ $$\int \dfrac{\sqrt{1-y^4}}{\sqrt{1-y^4}-y^2}dy$$

$$\int \dfrac{\sqrt{\dfrac{1}{y^4}-1}}{\sqrt{\dfrac{1}{y^4}-1}-1}dy$$

How to proceed from here or feel free to suggest shorter and clean approach

$\endgroup$
5
  • 4
    $\begingroup$ Is it $$\sqrt{\sqrt{1+x^4}-x^2}$$ $\endgroup$ Dec 26 '19 at 7:27
  • $\begingroup$ no,its the same I wrote $\endgroup$ Dec 26 '19 at 7:33
  • 5
    $\begingroup$ The reason we ask is because $\sqrt{\sqrt{x}}$ is an atrocious notation that could've easily been simplified to $\sqrt[4]{x}$ so it seems suspicious that it would be there like that. Second, the square root over both makes it so that the powers match up with the $x^2$. Lastly, this integral has no elementary antiderivative, but the altered version does. $\endgroup$ Dec 26 '19 at 7:39
  • $\begingroup$ Mathematica+Rubi can.Solution very large with Elliptic function and another elementary stuff. $\endgroup$ Dec 26 '19 at 10:48
  • $\begingroup$ shouldn't the second-to-last line of your attempt have fourth-roots instead of square-roots, i.e., $\int \dfrac{\sqrt[4]{1-y^4}}{\sqrt[4]{1-y^4}-y^2}dy$ ? $\endgroup$
    – David H
    Dec 28 '19 at 8:16
1
$\begingroup$

In the same spirit as Mariusz Iwaniuk's comment, a CAS gives $$\int \dfrac{\sqrt{1-y^4}}{\sqrt{1-y^4}-y^2}dy=\frac{1}{3} y^3 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};y^4,2 y^4\right)+\frac{1}{8} \left(4 y+2^{3/4} \left(\tan ^{-1}\left(\sqrt[4]{2} y\right)+\tanh ^{-1}\left(\sqrt[4]{2} y\right)\right)\right)$$ where appears the Appell hypergeometric function of two variables.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.