Find the ratio of integrals $\int_0^1 (1\pm x^4)^{-1/2}\,dx$

Question

How to find this ratio

$$\frac{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1+x^{4}}}\mathrm{d}x}{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1-x^{4}}}\mathrm{d}x}$$

without evaluating each integral? The integrals themselves can be expressed as elliptic integrals, but the ratio may be simpler.

Answer 1

For the numerator, change variable to $t = \tan\frac{\theta}{2}$, we get $$\int_0^1 \frac{dt}{\sqrt{1+t^4}} = \int_0^1 \frac{dt}{\sqrt{(1+t^2)^4 - 2t^2}} = \frac12\int_0^1 \frac{1}{\sqrt{1 - \frac12\left(\frac{2t}{1+t^2}\right)}}\frac{2dt}{1+t^2}\\ = \frac12 \int_0^{\pi/2}\frac{d\theta}{\sqrt{1-\frac12\sin^2\theta}} = \frac12 K\left(\frac{1}{\sqrt{2}}\right) $$ For the denominator, change variable to $t = \cos\theta$, we have

$$\int_0^1 \frac{dt}{\sqrt{1-t^4}} = \int_{\pi/2}^0\frac{d\cos\theta}{\sqrt{(1-\cos^2\theta)(1+\cos^2\theta)}} = \int_0^{\pi/2}\frac{d\theta}{\sqrt{1+\cos^2\theta}}\\ = \int_0^{\pi/2}\frac{d\theta}{\sqrt{2-\sin^2\theta}} = \frac{1}{\sqrt{2}}K\left(\frac{1}{\sqrt{2}}\right) $$ where $$K(k) = \int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}} = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}$$ is the complete elliptic integral of the first kind.

Compare the two expressions, it is clear the desired ratio is $\displaystyle\;\frac{1}{\sqrt{2}}$.

Update

Thinking more about this, we can completely get rid of the use of elliptic integral.

For the denominator, if we change variable to $t = \frac{1-s^2}{1+s^2}$, it becomes

$$ \int_1^0 \frac{d\left(\frac{1-s^2}{1+s^2}\right)}{\sqrt{1 - \left(\frac{1-s^2}{1+s^2}\right)^4}} = 4\int_0^1 \frac{sds}{\sqrt{(1+s^2)^4 - (1-s^2)^4}} = 4\int_0^1 \frac{sds}{\sqrt{8s^2 + 8s^6}} = \sqrt{2}\int_0^1 \frac{ds}{\sqrt{1+s^4}}$$ This is nothing but $\sqrt{2}$ times the numerator!

Answer 2

The first integral is a special case of the complete Elliptic integral of the first kind (see 1) $$I_1 = \int_0^1\frac{dx}{\sqrt{1+x^4}} = \frac{1}{2}K\left(\frac{\sqrt{2}}{2}\right) \approx 0.92703733865$$ and the second integral $$I_2 = \int_0^1\frac{dx}{\sqrt{1-x^4}} = \frac{1}{2}\varpi \approx 1.311028777$$ is related to the Lemniscate constant (see 2 or 3).

Both integrals be expressed in terms of the $\Gamma$ function $$I_1 = 2\frac{\Gamma(5/4)^2}{\sqrt{\pi}}, \quad I_2 = \frac{\Gamma(1/4)^2}{4\sqrt{2\pi}}$$ and their quotient is $\frac{\sqrt{2}}{2}$.