Evaluate $$\int_{0}^{1}\frac{1-x}{1+x}\frac{dx}{\sqrt{x+x^2+x^3}}$$

I think none of the properties of definite integral will be useful here so I think I will have to integrate. But I am unable to do so. Some hints on how to integrate. Thanks.

  • $\begingroup$ Numerical integration yields 1.0472... which you might use as a check on any symbolic solution. $\endgroup$ – David G. Stork May 26 '16 at 16:28
  • 1
    $\begingroup$ Looks like $\pi/3$ to me. $\endgroup$ – achille hui May 26 '16 at 21:21

Let $u = \sqrt{x+1+x^{-1}}$, we have $2 u du = (1 - x^{-2})dx$ and

$$ \int_0^1 \frac{1-x}{1+x}\frac{dx}{\sqrt{x+x^2+x^3}} = \int_\infty^\sqrt{3} \frac{1-x}{1+x}\frac{2u du}{(x-x^{-1})u} = 2 \int_\sqrt{3}^\infty \frac{du}{x+2+x^{-1}}\\ = 2 \int_\sqrt{3}^\infty \frac{du}{u^2+1} = 2 \left[\tan^{-1}u\right]_{\sqrt{3}}^\infty = 2 \left(\frac{\pi}{2} - \frac{\pi}{3}\right) = \frac{\pi}{3} $$


Mathematica gives the indefinite integral:

$\int {1-x \over 1+x}{1 \over \sqrt{x+x^2+x^3}} dx =$

$-\frac{2 \sqrt[6]{-1} x^{3/2} \left(i \sqrt{1-\frac{(-1)^{2/3}}{x}} \sqrt{\frac{x+\sqrt[3]{-1}}{x}} F\left(i \sinh ^{-1}\left(\frac{(-1)^{5/6}}{\sqrt{x}}\right)|(-1)^{2/3}\right)+\frac{2 \left(1+\sqrt[3]{-1}\right) \sqrt{\frac{\sqrt{x}-\sqrt[3]{-1}+1}{\left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1} \sqrt{x}-1\right)}} \sqrt{\frac{(-1)^{2/3} \sqrt{x}-1}{\left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1} \sqrt{x}-1\right)}} \sqrt{-\frac{(-1)^{2/3} \sqrt{x}+1}{\sqrt{x}+\sqrt[3]{-1}-1}} \left(\sqrt[3]{-1} \sqrt{x}-1\right)^2 \left(\left(\sqrt[3]{-1}+i\right) F\left(\left.\sin ^{-1}\left(\sqrt{\frac{\sqrt{x}-\sqrt[3]{-1}+1}{\left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1} \sqrt{x}-1\right)}}\right)\right|-3\right)-2 \sqrt[3]{-1} \Pi \left(\frac{-3 i+(1+2 i) \sqrt{3}}{(2+i)+\sqrt{3}};\left.\sin ^{-1}\left(\sqrt{\frac{\sqrt{x}-\sqrt[3]{-1}+1}{\left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1} \sqrt{x}-1\right)}}\right)\right|-3\right)\right)}{x}-\frac{2 \left(1+\sqrt[3]{-1}\right) \sqrt{\frac{\sqrt{x}-\sqrt[3]{-1}+1}{\left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1} \sqrt{x}-1\right)}} \sqrt{\frac{(-1)^{2/3} \sqrt{x}-1}{\left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1} \sqrt{x}-1\right)}} \sqrt{-\frac{(-1)^{2/3} \sqrt{x}+1}{\sqrt{x}+\sqrt[3]{-1}-1}} \left(\sqrt[3]{-1} \sqrt{x}-1\right)^2 \left(\left(\sqrt[3]{-1}-i\right) F\left(\left.\sin ^{-1}\left(\sqrt{\frac{\sqrt{x}-\sqrt[3]{-1}+1}{\left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1} \sqrt{x}-1\right)}}\right)\right|-3\right)-2 \sqrt[3]{-1} \Pi \left(-\frac{i \left(3+(2+i) \sqrt{3}\right)}{(-2+i)+\sqrt{3}};\left.\sin ^{-1}\left(\sqrt{\frac{\sqrt{x}-\sqrt[3]{-1}+1}{\left(1+\sqrt[3]{-1}\right) \left(\sqrt[3]{-1} \sqrt{x}-1\right)}}\right)\right|-3\right)\right)}{x}\right)}{\sqrt{x \left(x^2+x+1\right)}}$

which very strongly suggests that there's no simple method "by hand."

You can also do a Taylor series:

$\frac{1-x}{(x+1) \sqrt{x^3+x^2+x}} \approx \frac{1}{\sqrt{x}}-\frac{5 \sqrt{x}}{2}+\frac{23 x^{3/2}}{8}-\frac{37 x^{5/2}}{16}+\frac{203 x^{7/2}}{128}-\frac{355 x^{9/2}}{256}+O\left(x^{11/2}\right)$

and then integrate each term, but that will likely not give an adequate solution.

  • $\begingroup$ the result can not expressed by a known elementary function. $\endgroup$ – Dr. Sonnhard Graubner May 26 '16 at 16:33
  • $\begingroup$ Does Mathematica give a result for the definite integral? I doubt the result will be elementary, but the presence of the elliptic integrals in that answer makes me hope that the final result can be expressed using complete elliptic integrals. $\endgroup$ – Semiclassical May 26 '16 at 16:35
  • $\begingroup$ Okay, so how do i proceed? $\endgroup$ – user167045 May 26 '16 at 16:35
  • $\begingroup$ Use computer algebra. (I don't know why anyone is doing this kind of work "by hand" anymore, just as I don't know why anyone would multiply two 20-digit numbers by hand now that there are calculators.) $\endgroup$ – David G. Stork May 26 '16 at 16:46

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