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I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this:

$\hspace{1in}$The integrand graph

The approximate numeric value of the integral: $$I\approx8.372211626601275661625747121...$$

Neither Mathematica nor Maple could find a closed form for this integral, and lookups of the approximate numeric value in WolframAlpha and ISC+ did not return plausible closed form candidates either. But I still hope there might be a closed form for it.

I am also interested in cases when only numerator or only denominator is present under the logarithm.

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    $\begingroup$ Do you have any reason to believe there is a closed form for that horrid-looking thing? $\endgroup$ – dfeuer Nov 11 '13 at 17:12
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    $\begingroup$ In the meantime, I have been able to manipulate the integral into the following form: $$8 \int_0^{\infty} du \frac{(u^2-1)(u^4-6 u^2+1)}{u^8+4 u^6+70 u^4+4 u^2+1} \log{u}$$ from which I may deduce that there is in fact a closed form (because the roots of the denominator are expressible in closed form, a little messy but not bad). But because there are eight roots, a residue-based approach will be very very messy and almost hopeless without some computer algebra. $\endgroup$ – Ron Gordon Nov 12 '13 at 0:21
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    $\begingroup$ A related problem. $\endgroup$ – Mhenni Benghorbal Nov 19 '13 at 18:46
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    $\begingroup$ @MhenniBenghorbal: Related how? $\endgroup$ – Ron Gordon Nov 19 '13 at 21:02
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    $\begingroup$ Out of curiosity, is there an application for this integral? $\endgroup$ – linuxfreebird Nov 16 '14 at 0:35
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I will transform the integral via a substitution, break it up into two pieces and recombine, perform an integration by parts, and perform another substitution to get an integral to which I know a closed form exists. From there, I use a method I know to attack the integral, but in an unusual way because of the 8th degree polynomial in the denominator of the integrand.

First sub $t=(1-x)/(1+x)$, $dt=-2/(1+x)^2 dx$ to get

$$2 \int_0^{\infty} dt \frac{t^{-1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} $$

Now use the symmetry from the map $t \mapsto 1/t$. Break the integral up into two as follows:

\begin{align} & 2 \int_0^{1} dt \frac{t^{-1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} + 2 \int_1^{\infty} dt \frac{t^{-1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} \\ &= 2 \int_0^{1} dt \frac{t^{-1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} + 2 \int_0^{1} dt \frac{t^{1/2}}{1-t^2} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} \\ &= 2 \int_0^{1} dt \frac{t^{-1/2}}{1-t} \log{\left (\frac{5-2 t+t^2}{1-2 t +5 t^2} \right )} \end{align}

Sub $t=u^2$ to get

$$4 \int_0^{1} \frac{du}{1-u^2} \log{\left (\frac{5-2 u^2+u^4}{1-2 u^2 +5 u^4} \right )}$$

Integrate by parts:

$$\left [2 \log{\left (\frac{1+u}{1-u} \right )} \log{\left (\frac{5-2 u^2+u^4}{1-2 u^2 +5 u^4} \right )}\right ]_0^1 \\- 32 \int_0^1 du \frac{\left(u^5-6 u^3+u\right)}{\left(u^4-2 u^2+5\right) \left(5 u^4-2 u^2+1\right)} \log{\left (\frac{1+u}{1-u} \right )}$$

One last sub: $u=(v-1)/(v+1)$ $du=2/(v+1)^2 dv$, and finally get

$$8 \int_0^{\infty} dv \frac{(v^2-1)(v^4-6 v^2+1)}{v^8+4 v^6+70v^4+4 v^2+1} \log{v}$$

With this form, we may finally conclude that a closed form exists and apply the residue theorem to obtain it. To wit, consider the following contour integral:

$$\oint_C dz \frac{8 (z^2-1)(z^4-6 z^2+1)}{z^8+4 z^6+70z^4+4 z^2+1} \log^2{z}$$

where $C$ is a keyhole contour about the positive real axis. This contour integral is equal to (I omit the steps where I show the integral vanishes about the circular arcs)

$$-i 4 \pi \int_0^{\infty} dv \frac{8 (v^2-1)(v^4-6 v^2+1)}{v^8+4 v^6+70v^4+4 v^2+1} \log{v} + 4 \pi^2 \int_0^{\infty} dv \frac{8 (v^2-1)(v^4-6 v^2+1)}{v^8+4 v^6+70v^4+4 v^2+1}$$

It should be noted that the second integral vanishes; this may be easily seen by exploiting the symmetry about $v \mapsto 1/v$.

On the other hand, the contour integral is $i 2 \pi$ times the sum of the residues about the poles of the integrand. In general, this requires us to find the zeroes of the eight degree polynomial, which may not be possible analytically. Here, on the other hand, we have many symmetries to exploit, e.g., if $a$ is a root, then $1/a$ is a root, $-a$ is a root, and $\bar{a}$ is a root. For example, we may deduce that

$$z^8+4 z^6+70z^4+4 z^2+1 = (z^4+4 z^3+10 z^2+4 z+1) (z^4-4 z^3+10 z^2-4 z+1)$$

which exploits the $a \mapsto -a$ symmetry. Now write

$$z^4+4 z^3+10 z^2+4 z+1 = (z-a)(z-\bar{a})\left (z-\frac{1}{a}\right )\left (z-\frac{1}{\bar{a}}\right )$$

Write $a=r e^{i \theta}$ and get the following equations:

$$\left ( r+\frac{1}{r}\right ) \cos{\theta}=-2$$ $$\left (r^2+\frac{1}{r^2}\right) + 4 \cos^2{\theta}=10$$

From these equations, one may deduce that a solution is $r=\phi+\sqrt{\phi}$ and $\cos{\theta}=1/\phi$, where $\phi=(1+\sqrt{5})/2$ is the golden ratio. Thus the poles take the form

$$z_k = \pm \left (\phi\pm\sqrt{\phi}\right) e^{\pm i \arctan{\sqrt{\phi}}}$$

Now we have to find the residues of the integrand at these 8 poles. We can break this task up by computing:

$$\sum_{k=1}^8 \operatorname*{Res}_{z=z_k} \left [\frac{8 (z^2-1)(z^4-6 z^2+1) \log^2{z}}{z^8+4 z^6+70z^4+4 z^2+1}\right ]=\sum_{k=1}^8 \operatorname*{Res}_{z=z_k} \left [\frac{8 (z^2-1)(z^4-6 z^2+1)}{z^8+4 z^6+70z^4+4 z^2+1}\right ] \log^2{z_k}$$

Here things got very messy, but the result is rather unbelievably simple:

$$\operatorname*{Res}_{z=z_k} \left [\frac{8 (z^2-1)(z^4-6 z^2+1)}{z^8+4 z^6+70z^4+4 z^2+1}\right ] = \text{sgn}[\cos{(\arg{z_k})}]$$

EDIT

Actually, this is a very simple computation. Inspired by @sos440, one may express the rational function of $z$ in a very simple form:

$$\frac{8 (z^2-1)(z^4-6 z^2+1)}{z^8+4 z^6+70z^4+4 z^2+1} = -\left [\frac{p'(z)}{p(z)} + \frac{p'(-z)}{p(-z)} \right ]$$

where

$$p(z)=z^4+4 z^3+10 z^2+4 z+1$$

The residue of this function at the poles are then easily seen to be $\pm 1$ according to whether the pole is a zero of $p(z)$ or $p(-z)$.

END EDIT

That is, if the pole has a positive real part, the residue of the fraction is $+1$; if it has a negative real part, the residue is $-1$.

Now consider the log piece. Expanding the square, we get 3 terms:

$$\log^2{|z_k|} - (\arg{z_k})^2 + i 2 \log{|z_k|} \arg{z_k}$$

Summing over the residues, we find that because of the $\pm1$ contributions above, that the first and third terms sum to zero. This leaves the second term. For this, it is crucial that we get the arguments right, as $\arg{z_k} \in [0,2 \pi)$. Thus, we have

$$\begin{align}I= \int_0^{\infty} dv \frac{8 (v^2-1)(v^4-6 v^2+1)}{v^8+4 v^6+70v^4+4 v^2+1} \log{v} &= \frac12 \sum_{k=1}^8 \text{sgn}[\cos{(\arg{z_k})}] (\arg{z_k})^2 \\ &= \frac12 [2 (\arctan{\sqrt{\phi}})^2 + 2 (2 \pi - \arctan{\sqrt{\phi}})^2 \\ &- 2 (\pi - \arctan{\sqrt{\phi}})^2 - 2 (\pi + \arctan{\sqrt{\phi}})^2]\\ &= 2 \pi^2 -4 \pi \arctan{\sqrt{\phi}} \\ &= 4 \pi \, \text{arccot}{\sqrt{\phi}}\\\end{align}$$

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    $\begingroup$ +1. $\text{arccot}(\sqrt{\phi})$-definitely one of the most weirdest closed form solutions to have ever been obtained! $\endgroup$ – user17762 Dec 2 '13 at 6:31
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    $\begingroup$ @ShikariShambu: which makes me wonder if we can generate a list of the oddest closed-form solutions to integrals, sums, products, diff eq'ns, and the like. What makes a closed form "odd"? What makes this one odd? The juxtaposition of the arccotangent and $\phi$? This could lead to...a weird discussion, but a fun one nonetheless. $\endgroup$ – Ron Gordon Dec 18 '13 at 23:41
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    $\begingroup$ I know that $\phi$ has some nice properties, but at the end of the day it's just another algebraic number: $$\phi = \frac{1+\sqrt{5}}{2},$$ and all we've done is taken the arc cotangent of the square root of it and multiplied by $4\pi$. - just wondering why this result would be seen as "odd". It most likely appears because it happens to be a root of some polynomial which has $(1+\sqrt{5})/2$ as one of its roots. $\endgroup$ – Pixel Feb 12 '14 at 18:11
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    $\begingroup$ @RonGordon When I see things like this, I don't consider it odd so much as a pleasant surprise. Phi is known as a number of beauty for many reasons, and to see it emerge from, to quote our friend dfeuer, "that horrid-looking thing", is quite a pleasant surprise indeed! No tin-foil hats, no cube-root-of-31-conspiracies, just running into an old friend at the grocery =) $\endgroup$ – corsiKa Mar 13 '14 at 19:28
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    $\begingroup$ @corsiKa: $\phi$ alone, sure. But now imagine your meeting your old friend, but dressed in drag with a Kaiser-era military helmet on, spike and all. That's sort of the feeling you get when you see, not regular old $\phi$, but ARCCOT SQRT $\phi$. $\endgroup$ – Ron Gordon Mar 14 '14 at 17:08
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NEW ANSWER. I found yet another way of solving this problem. My new solution does not use contour integration, and is based on the following observation: for $|z| \leq 1$,

$$ - \int_{-1}^{1} \frac{1}{x} \sqrt{\frac{1+x}{1-x}} \log(1 - zx) \, dz= \pi \sin^{-1} z - \pi \log \left( \tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z^{2}} \right) . $$

As I want to keep both the old answer and the new answer, I posted my new solution to other page. You can check it here.


OLD ANSWER. Okay here is another solution. It is also related to my generalization.

We claim the following proposition:

Proposition. If $0 < r < 1$ and $r < s$, then $$ I(r, s) := \int_{-1}^{1} \frac{1}{x} \sqrt{\frac{1+x}{1-x}} \log \left( \frac{1 + 2rsx + (r^{2} + s^{2} - 1)x^{2}}{1 - 2rsx + (r^{2} + s^{2} - 1)x^{2}} \right) \, dx = 4\pi \arcsin r. \tag{1} $$

Assuming this proposition, all that we have to do is to solve the non-linear system of equations

$$ 2rs = 2 \quad \text{and} \quad r^{2} + s^{2} - 1 = 2. $$

The unique solution satisfying the condition of the proposition is $r = \phi - 1$ and $s = \phi$. So by $\text{(1)}$ we have

\begin{align*} \int_{-1}^{1} \frac{1}{x} \sqrt{\frac{1+x}{1-x}} \log \left( \frac{1 + 2x + 2x^{2}}{1 - 2x + 2x^{2}} \right) \, dx & = I(\phi-1, \phi) \\ &= 4\pi \arcsin (\phi - 1) = 4\pi \operatorname{arccot} \sqrt{\phi}. \end{align*}

Thus it remains to prove the proposition.


Proof of Proposition. We divide the proof into several steps.

Step 1. (Case reduction by analytic continuation) We first remark that given $r$ and $s$, we always have

$$ \min_{-1 \leq x \leq 1} \{ 1 \pm 2rsx + (r^{2} + s^{2} - 1)x^{2} \} > 0. \tag{2} $$

Indeed, it is not hard to check if we utilize the following equality

$$ 1 \pm 2rsx + (r^{2} + s^{2} - 1)x^{2} = (1 \pm rsx)^{2} - (1 - r^{2})(1 - s^{2}) x^{2}. $$

Then $\text{(2)}$ shows that the integrand of $I(r, s)$ remains holomoprhic under small perturbation of $s$ in $\Bbb{C}$. So it allows us to extend $s \mapsto I(r, s)$ as a holomorphic function on some open set containing the line segment $(r, \infty) \subset \Bbb{C}$. Then by the principle of analytic continuation, it is sufficient to prove that $\text{(1)}$ holds for $r < s < 1$.

Step 2. (Integral representation of $I$) Put $r = \sin \alpha$ and $s = \sin \beta$, where $ 0 < \alpha < \beta < \frac{\pi}{2}$. Then

\begin{align*} I(r, s) &= \int_{-1}^{1} \frac{1+x}{x\sqrt{1-x^{2}}} \log \left( \frac{1 + 2rsx + (r^{2} + s^{2} - 1)x^{2}}{1 - 2rsx + (r^{2} + s^{2} - 1)x^{2}} \right) \, dx \\ &= \int_{0}^{1} \frac{2}{x\sqrt{1-x^{2}}} \log \left( \frac{1 + 2rsx + (r^{2} + s^{2} - 1)x^{2}}{1 - 2rsx + (r^{2} + s^{2} - 1)x^{2}} \right) \, dx \qquad (\because \text{ parity}) \\ &= \int_{1}^{\infty} \frac{2}{\sqrt{x^{2}-1}} \log \left( \frac{x^{2} + 2rsx + (r^{2} + s^{2} - 1)}{x^{2} - 2rsx + (r^{2} + s^{2} - 1)} \right) \, dx \qquad (x \mapsto x^{-1}) \\ &= \int_{0}^{1} \frac{2}{t} \log \left( \frac{\left(t+t^{-1}\right)^{2} + 4rs\left(t+t^{-1}\right) + 4(r^{2} + s^{2} - 1)}{\left(t+t^{-1}\right)^{2} - 4rs\left(t+t^{-1}\right) + 4(r^{2} + s^{2} - 1)} \right) \, dt, \end{align*}

where in the last line we utilized the substitution $x = \frac{1}{2}(t + t^{-1})$. If we introduce the quartic polynomial \begin{align*} p(t) = t^{4} + 4rst^{3} + (4r^{2}+4s^{2}-2)t^{2} + 4rst + 1, \end{align*}

then by the property $p(1/t) = t^{-4}p(t)$, we can simplify

\begin{align*} I(r, s) &= 2 \int_{0}^{1} \frac{\log p(t) - \log p(-t)}{t} \, dt = \int_{0}^{\infty} \frac{\log p(t) - \log p(-t)}{t} \, dt \\ &= - \int_{0}^{\infty} \left( \frac{p'(t)}{p(t)} + \frac{p'(-t)}{p(-t)} \right) \log t \, dt = - \frac{1}{2} \Re \int_{-\infty}^{\infty} \left( \frac{p'(z)}{p(z)} + \frac{p'(-z)}{p(-z)} \right) \log z \, dz, \end{align*}

where we choose the branch cut of $\log$ in such a way that it avoids the upper-half plane

$$\Bbb{H} = \{ z \in \Bbb{C} : \Im z > 0 \}.$$

Step 3. (Residue calculation) Since

$$ f(z) := \left( \frac{p'(z)}{p(z)} + \frac{p'(-z)}{p(-z)} \right) \log z = O\left(\frac{\log z}{z^{2}} \right) \quad \text{as } z \to \infty, $$

by replacing the contour of integration by a semicircle of sufficiently large radius, it follows that

\begin{align*} I(r, s) = - \frac{1}{2} \Re \left\{ 2 \pi i \sum_{z_{0} \in \Bbb{H}} \operatorname{Res}_{z = z_{0}} f(z) \right\} = \pi \Im \sum_{z_{0} \in \Bbb{H}} \operatorname{Res}_{z = z_{0}} f(z). \end{align*}

(It turns out that $f(z)$ has only logarithmic singularity at the origin. So it does not account for the value of $I(r, s)$.) But by a simple calculation, together with the condition $ 0 < \alpha < \beta < \frac{\pi}{2}$, we easily notice that the zeros of $p(z)$ are exactly

$$ e^{\pm i(\alpha + \beta)} \quad \text{and} \quad -e^{\pm i(\alpha - \beta)}. $$

Now let $Z_{+}$ be the set of zeros of $p(z)$ in $\Bbb{H}$ and $Z_{-}$ be the set of zeros of $p(z)$ in $-\Bbb{H}$. Then

$$ Z_{+} = \{ e^{i(\beta+\alpha)}, -e^{-i(\beta - \alpha)} \} \quad \text{and} \quad Z_{-} = \{ e^{-i(\beta+\alpha)}, -e^{i(\beta- \alpha)} \}. $$

This in particular shows that

$$ \frac{p'(z)}{p(z)}\log z = \sum_{z_{0} \in Z_{+}} \frac{\log z}{z - z_{0}} + \text{holomorphic function on } \Bbb{H} $$

and

$$ \frac{p'(-z)}{p(-z)}\log z = -\sum_{z_{0} \in -Z_{-}} \frac{\log z}{z - z_{0}} + \text{holomorphic function on } \Bbb{H}. $$

So it follows that

\begin{align*} I(r, s) &= \pi \Im \left\{ \sum_{z_{0} \in Z_{+}} \log z_{0} - \sum_{z_{0} \in -Z_{-}} \log z_{0} \right\} \\ &= \pi \Im \left\{ \log e^{i(\beta+\alpha)} + \log e^{i(\pi-\beta+\alpha)} - \log e^{i(\pi-\beta-\alpha)} - \log e^{i(\beta-\alpha)} \right\} \\ &= \pi \Im \left\{ i(\beta+\alpha) + i(\pi-\beta+\alpha) - i(\pi-\beta-\alpha) - i(\beta-\alpha) \right\} \\ &= 4\pi \alpha = 4\pi \arcsin r. \end{align*}

This completes the proof.

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    $\begingroup$ Very nice, and it helps me simplify a piece of my proof as well (I think our solutions have more in common than not). One question, though: what about the branch point of the log in the residue calculation? I know it doesn't seem to matter as you do end up with the correct solution, but you may want to say something about avoiding the branch point at the origin and defining a branch of the log (which I think you do anyway with your restrictions on $\alpha$ and $\beta$) in the complex plane. $\endgroup$ – Ron Gordon Nov 17 '13 at 8:45
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    $\begingroup$ @RonGordon, As written in the solution, the branch cut of the log is chosen so that it avoids the upper half plane. So it would be safe if we choose it as the negative y-axis, but I think the standard branch cut $(-\infty, 0)$ would also works. $\endgroup$ – Sangchul Lee Nov 17 '13 at 15:44
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    $\begingroup$ Oh my, there it is. My bad, so sorry. In any case, reading your solution helped me simplify a small part of mine, so thanks. $\endgroup$ – Ron Gordon Nov 17 '13 at 15:53
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$\large\hspace{3in}I=4\,\pi\operatorname{arccot}$$\sqrt\phi$

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    $\begingroup$ @OliverBel IMHO, if the question is not asking explicitly for a proof, and there is no obvious conjectured closed form, I think it's OK to post the result first and add a proof later, when time permits. The result itself can be useful for the person asking. It may be downvoted if not useful, but it still qualifies as an answer. The guideline for comments say "Avoid answering questions in comments", and I see no reason to withhold the result until the full proof is ready to be posted. $\endgroup$ – Vladimir Reshetnikov Nov 11 '13 at 22:20
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    $\begingroup$ @VladimirReshetnikov: I defer to Hamming: "The purpose of computing is insight, not numbers." Unless the result itself is particularly illuminating, I do not agree that it is an answer. There is a user who continues to insist on putting Maple output as an answer, and I think that degrades the site. I'd prefer if Cleo gave some insight as to where this answer came from. $\endgroup$ – Ron Gordon Nov 11 '13 at 22:53
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    $\begingroup$ @RonGordon Of course, I would prefer to see the proof too. I hope that Cleo posts it eventually, or somebody else does it, inspired by the fact the closed form exists (and maybe also guided by its shape). My point is that writing a proof clearly and typesetting it can take hours (at least for me) and I can imagine that not everybody can allocate the required time promptly. I would prefer to at least see the result posted promptly in such cases. (And from my experience being a PhD student in theoretical physics I can say that sometimes the result is the only thing a person cares of :) $\endgroup$ – Vladimir Reshetnikov Nov 11 '13 at 23:17
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    $\begingroup$ This style of answer is complete disrespect. This situation seems for me like this: Cleo found interesting problem, and solved it. He is lazy to write the solution but want to show how clever he is, so decided to post only the final result. The reference to the definition of golden ratio made me laugh. If OP asks question of such level he definitely familiar with this constant. Note that this is not a single example. ALL Cleo's answer are of this style, and even after polite ephasis that these answers is not what OP's wanted he continues to post only final results! $\endgroup$ – Norbert Nov 17 '13 at 9:14
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    $\begingroup$ «Disrespectful answer» and «arrogant answer»? Really?! I find the answer pretty useless, as I did not learn anything from it, but disrespectful and arrogant are judgements which seem totally inappropriate to this answer! I really wish commenters would limit the dramatic charge of their comments to ¡the value of an integral! $\endgroup$ – Mariano Suárez-Álvarez Apr 26 '14 at 2:16
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Our aim is to give an elementary proof of Proposition formula (1) in the answer of @sos440. We first note that $$ \min_{-1\leq x\leq1}\{1\pm2rsx+(r^{2}+s^{2}-1)x^{2}\}>0. $$ Indeed, if $x=\pm1$ then $$ 1\pm2rsx+(r^{2}+s^{2}-1)x^{2}\geq(r-s)^{2}>0, $$ if $x=0$ then $$ 1\pm2rsx+(r^{2}+s^{2}-1)x^{2}=1>0, $$ if $-1<x<1$, $x\neq0$ then the equations \begin{eqnarray*} \frac{\partial}{\partial s}(1\pm2rsx+(r^{2}+s^{2}-1)x^{2}) & = & 0,\\ \frac{\partial}{\partial r}(1\pm2rsx+(r^{2}+s^{2}-1)x^{2}) & = & 0, \end{eqnarray*} give $\pm r=sx$, $\pm s=rx$, which is impossible.

In the second step we show that $I(r,s)$ is independent of $s$. $$ \frac{\partial}{\partial s}I(r,s)=\int_{-1}^{1}\sqrt{\frac{1+x}{1-x}}\cdot\frac{4r(1+(r^{2}-s^{2}-1)x^{2})}{(1-2rsx+(r^{2}+s^{2}-1)x^{2})(1+2rsx+(r^{2}+s^{2}-1)x^{2}}\, dx. $$ Substituting $x:=-x$ and adding them we obtain $$ 2\frac{\partial}{\partial s}I(r,s)=\int_{-1}^{1}\frac{2}{\sqrt{1-x^{2}}}\cdot\frac{4r(1+(r^{2}-s^{2}-1)x^{2})}{(1-2rsx+(r^{2}+s^{2}-1)x^{2})(1+2rsx+(r^{2}+s^{2}-1)x^{2}}\, dx, $$ that is, $$ \frac{\partial}{\partial s}I(r,s)=\int_{-1}^{1}\frac{1}{\sqrt{1-x^{2}}}\cdot\frac{4r(-s^{2}+r^{2}-1)x^{2}+4r}{1+(r^{2}+s^{2}-1)^{2}x^{4}+(2s^{2}-4r^{2}s^{2}+2r^{2}-2)x^{2}}\, dx. $$ Substituting $x:=\sin(t)$ we have $$ \frac{\partial}{\partial s}I(r,s) = \int_{-\pi/2}^{\pi/2}\frac{4r(-s^{2}+r^{2}-1)\sin(t)^{2}+4r}{1+(r^{2}+s^{2}-1)^{2}\sin(t)^{4}+(2s^{2}-4r^{2}s^{2}+2r^{2}-2)\sin(t)^{2}}\, dt $$ $$ =\int_{-\pi/2}^{\pi/2}-\frac{8r((-s^{2}+r^{2}-1)\cos(2t)+s^{2}-r^{2}-1)}{(r^{2}+s^{2}-1)^{2}\cos(2t)^{2}-2(r^{2}-s^{2}-1)(r^{2}+1-s^{2})\cos(2t)+r^{4}+(2-6s^{2})r^{2}+(s^{2}+1)^{2}}\, dt $$ $$ = \int_{-\pi}^{\pi}-\frac{4r((-s^{2}+r^{2}-1)\cos(y)+s^{2}-r^{2}-1)}{(r^{2}+s^{2}-1)^{2}\cos(y)^{2}-2(r^{2}-s^{2}-1)(r^{2}+1-s^{2})\cos(y)+r^{4}+(2-6s^{2})r^{2}+(s^{2}+1)^{2}}\, dy. $$ Introducing the new variable $T:=\tan\frac{y}{2}$ we obtain \begin{eqnarray*} \frac{\partial}{\partial s}I(r,s) & = & \int_{-\infty}^{\infty}-\frac{4r(s^{2}-r^{2})T^{2}-4r}{(r-s)^{2}(r+s)^{2}T^{4}+((2-4s^{2})r^{2}+2s^{2})T^{2}+1}\, dT\\ & = & -\frac{4r(s^{2}-r^{2})}{(r-s)^{2}(r+s)^{2}}\int_{-\infty}^{\infty}\frac{T^{2}+a}{T^{4}+bT^{2}+b^{2}/4+d}\, dT\\ & = & -\frac{4r(-s^{2}+r^{2})}{(r-s)^{2}(r+s)^{2}}\cdot\frac{(2a(b^{2}+4d)+(b^{2}+4d)^{3/2})\pi}{(b^{2}+4d)^{3/2}\sqrt{\sqrt{b^{2}+4d}+b}}, \end{eqnarray*} where $$ a=-\frac{1}{s^{2}-r^{2}}, $$ $$ b=\frac{(2-4s^{2})r^{2}+2s^{2}}{(r-s)^{2}(r+s)^{2}}, $$ $$ b^{2}+4d=\frac{4}{(r-s)^{2}(r+s)^{2}}. $$ It gives $2ab^{2}+8da+(b^{2}+4d)^{3/2}=0$.

Since $\frac{\partial}{\partial s}I(r,s)=0$ we have $$ I(r,s)=I(r,1)=\int_{-1}^{1}\frac{1}{x}\sqrt{\frac{1+x}{1-x}}\log\left(\frac{(1+rx)^{2}}{(1-rx)^{2}}\right)dx. $$ From this $$ \frac{\partial}{\partial r}I(r,1)=\int_{-1}^{1}\sqrt{\frac{1+x}{1-x}}\frac{4}{1-r^{2}x^{2}}\, dx. $$ Similarly as above we get $$ \frac{\partial}{\partial r}I(r,1)=\int_{-1}^{1}\frac{4}{\sqrt{1-x^{2}}(1-r^{2}x^{2})}\, dx=\frac{4\pi}{\sqrt{1-r^{2}}}=4\pi(\arcsin r)'. $$ It implies $$ I(r,1)=4\pi\arcsin r+C. $$ Taking the limit $\lim_{r\to0+}$ we obtain $C=0$, that is, $I(r,s)=4\pi\arcsin r$.

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    $\begingroup$ @sos440 Thanks. It was not a trivial task. How could you discover your general Proposition? $\endgroup$ – vesszabo Mar 9 '14 at 19:28
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    $\begingroup$ I first tried variants of the original integral by changing coefficients. Using inverse symbolic calculators, I found some patterns. Then I tried choose a nice parameters that makes the (conjectured) result look simple. $\endgroup$ – Sangchul Lee Mar 9 '14 at 20:22
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For the purposes of alternative methods, it may be of interest to note that the integrand

$$f(x)=\frac{1}{x}\sqrt{\frac{1+x}{1-x}}\log\left(\frac{2x^2+2x+1}{2x^2-2x+1}\right)$$ may be rewritten in terms of hyperbolic trigonometric functions. Using $$\tanh^{-1}(z) = \frac{1}{2}\log\left(\frac{1+z}{1-z}\right),$$ and we obtain

$$f(x)=\frac{1}{x}e^{\tanh^{-1}x}\log\left(\frac{1+\frac{2x}{1+2x^2}}{1-\frac{2x}{1+2x^2}}\right) = e^{\tanh^{-1} x}\left(\frac{2\tanh^{-1}\left(\frac{2x}{1+2x^2}\right)}{x}\right).$$

The rational function in the bracket, which we will denote $s(x)$, is symmetric about $x=0$.

The desired integral is

$$I=\int_{-1}^1 f(x)dx = \int_{-1}^1e^{\tanh^{-1}x}s(x)dx,$$

which, by adding the indicated useful definite integral to both side, gives

$$I + \int_{-1}^1 e^{-\tanh^{-1}x}s(x)dx = 2\int_{-1}^1 \frac{s(x)dx}{\sqrt{1-x^2}}.$$

Now using the change of variable $x=-y$ we have $$\int_{-1}^1 e^{-\tanh^{-1} x}s(x)dx = -\int_1^{-1} e^{\tanh y}s(-y)dy = \int_{-1}^1 e^{\tanh y}s(y)dy = I,$$ by the symmetry of $s(x)$. Hence, we finally obtain

$$I = \int_{-1}^1\frac{s(x)dx}{\sqrt{1-x^2}} = 2\int_{-1}^1\frac{1}{x\sqrt{1-x^2}}\tanh^{-1}\left(\frac{2x}{1+2x^2}\right)dx.$$

This integral is symmetric about $x=0$, so we have

$$I=4\int_0^1\frac{1}{x\sqrt{1-x^2}}\tanh^{-1}\left(\frac{2x}{1+2x^2}\right)dx,$$ which can be rewritten $$I=-4\int_0^1\left(\frac{d}{dx}\text{sech}^{-1}x\right)\tanh^{-1}\left(\frac{2x}{1+2x^2}\right)dx.$$

Using integration by parts this results in

$$I=8\int_0^1\frac{\text{sech}^{-1}(x)(1-2x^2)}{1+4x^4}dx.$$


We could also make the change of variable $y=\text{sech}^{-1}x$ to obtain

$$I=8\int_0^\infty\frac{y(\cosh^2(y)-2)\sinh y}{\cosh^4(y)+4}dy= 8\int_0^\infty\frac{y\sinh^3 y}{\cosh^4y+4}dy-8\int_0^\infty\frac{y\sinh y}{\cosh^4 y+4}dy.$$

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    $\begingroup$ How have I not seen this one prior 0_0 gorgeous solution $\endgroup$ – Brevan Ellefsen May 15 '17 at 22:54
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This answer provides a way to find $I=\displaystyle\int_0^1\dfrac{\ln\left(x^4-2x^2+5\right)-\ln\left(5x^4-2x^2+1\right)}{1-x^2}\ dx$ (which @RonGordon obtained above) with differentiating under the integral sign. A $u$-substitution of $u=\dfrac{1+x^2}{1-x^2}$ yields this.

$$I=\dfrac{1}{2}\displaystyle\int_1^\infty\dfrac{\ln\left(\frac{u^2+2u+2}{u^2-2u+2}\right)}{\sqrt{u^2-1}}\ du.$$ Now integrate by parts with $a=\ln\left(\frac{u^2+2u+2}{u^2-2u+2}\right)$ and $db=\dfrac{du}{\sqrt{u^2-1}}.$ $$I=\left.\ln\left(\dfrac{u^2+2u+2}{u^2-2u+2}\right)\ln(u+\sqrt{u^2-1})\right]^\infty_1+2\displaystyle\int_1^\infty\dfrac{u^2-2}{u^4+4}\ln\left(u+\sqrt{u^2-1}\right)\ du$$ The first term is equal to $0$, so we are left with this. $$I=2\displaystyle\int_1^\infty\dfrac{u^2-2}{u^4+4}\ln\left(u+\sqrt{u^2-1}\right)\ du$$ We now begin the step of differentiating under the integral. Consider the following integral: $$f(a)=a\displaystyle\int_1^\infty\dfrac{x^2-a^2}{x^4+a^4}\ln\left(x+\sqrt{x^2-1}\right)\ dx$$ Note that trivially, $f(0)=0.$ A quick $u=\dfrac{x}{a}$ yields this. $$f(a)=\displaystyle\int_{\frac{1}{a}}^\infty\dfrac{u^2-1}{u^4+1}\ln\left(au+\sqrt{(au)^2-1}\right)\ du$$ Differentiating with respect to $a$ and using the Chain Rule, we get this. $$f'(a)=-1\times\dfrac{-1}{a^2}\times\dfrac{\left(\frac{1}{a}\right)^2-1}{\left(\frac{1}{a}\right)^4+1}\ln\left(a\left(\dfrac{1}{a}\right)+\sqrt{\left(a\left(\dfrac{1}{a}\right)\right)^2-1}\right)+\displaystyle\int_{\frac{1}{a}}^\infty\dfrac{x^2-1}{x^4+1}\times\dfrac{x}{\sqrt{(ax)^2-1}}\ dx$$ Luckily, the first term cancels, so we are left with this. $$f'(a)=\displaystyle\int_{\frac{1}{a}}^\infty\dfrac{x^2-1}{x^4+1}\times\dfrac{x}{\sqrt{(ax)^2-1}}\ dx$$ A $u$-substitution of $u=\sqrt{(ax)^2-1}$ yields this. $$f'(a)=\displaystyle\int_0^\infty\dfrac{u^2+1-a^2}{(u^2+1)^2+a^4}\ du$$ Consider the integral with $u\mapsto\dfrac{\sqrt{a^4+1}}{u}$ $$f'(a)=\dfrac{1}{\sqrt{a^4+1}}\displaystyle\int_0^\infty\dfrac{(1-a^2)u^2+(a^4+1)}{u^4+2u^2+(a^2+1)}\ du$$ If we add these two versions of the integral and divided the numerator and denominator of the integrand by $u^2$, we get the following. $$f'(a)=\dfrac{(1-a^2)+\sqrt{a^4+1}}{2\sqrt{a^4+1}}\times\displaystyle\int_0^\infty\dfrac{1+\frac{\sqrt{a^4+1}}{u^2}}{\left(u-\frac{\sqrt{a^4+1}}{u}\right)^2+2\left(1+\sqrt{a^4+1}\right)}\ du$$ We can finally perform a very nice substitution of $w=u-\dfrac{\sqrt{a^4+1}}{u}$ to solve this integral. $$f'(a)=\dfrac{(1-a^2)+\sqrt{a^4+1}}{2\sqrt{a^4+1}}\times\displaystyle\int_{-\infty}^\infty\dfrac{dw}{w^2+2\left(1+\sqrt{a^4+1}\right)}\ dw$$ Thus, we can finally say that $f'(a)=\dfrac{(1-a^2)+\sqrt{a^4+1}}{2\sqrt{a^4+1}}\times\dfrac{\pi}{\sqrt{2\left(1+\sqrt{a^4+1}\right)}}.$ After a bit of considerable algebra, we can simply that to obtain this. $$f'(a)=\dfrac{\pi}{2}\sqrt{\dfrac{\sqrt{a^4+1}-a^2}{a^4+1}}$$ Integrating, we can now say this about the value of $f(a).$ $$f(a)=\dfrac{\pi}{2}\displaystyle\int_0^a\sqrt{\dfrac{\sqrt{x^4+1}-x^2}{x^4+1}}\ dx$$ Only one $u$-substitution of $u=\sqrt{x^4+1}-x^2$ is required here to obtain this. $$f(a)=\dfrac{\pi}{2\sqrt{2}}\displaystyle\int_{\sqrt{a^4+1}-a^2}^1\dfrac{du}{\sqrt{1-u^2}}$$ This, of course, is equal to $\dfrac{\pi\arccos\left(\sqrt{a^4+1}-a^2\right)}{2\sqrt{2}}.$

We will now manipulate this result to a function with $\arctan$ in it.

$f(a)=\dfrac{\pi\arccos\left(\sqrt{a^4+1}-a^2\right)}{2\sqrt{2}}=\dfrac{\pi}{\sqrt{2}}\arctan\left(\sqrt{\dfrac{\sqrt{a^4+1}-1}{a^2}}\right)$

Our desired value for our original integral is $\sqrt{2}f\left(\sqrt{2}\right).$

$$\boxed{\displaystyle\int_0^1\dfrac{\ln\left(x^4-2x^2+5\right)-\ln\left(5x^4-2x^2+1\right)}{1-x^2}\ dx=\pi\arctan\left(\sqrt{\dfrac{\sqrt{5}-1}{2}}\right)=\pi\text{arccot}\sqrt{\phi}}$$

So the final answer to the original problem is $4\pi\text{arccot}\sqrt{\phi}.$

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    $\begingroup$ Credit goes to Ronak's answer on Brilliant.org (Pulished on Dec 31, 2015). $\endgroup$ – MathGod Dec 27 '16 at 6:56
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    $\begingroup$ @IshanSingh I didn't copy from that answer; I was given the original integral and subsequently the initial $u=\dfrac{1+x^2}{1-x^2}$ by a friend who posed this question to me. $\endgroup$ – Arcturus Dec 28 '16 at 7:58
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Noteworthy, RIES (http://mrob.com/pub/ries/index.html) finds closed form from numerical value in the form of an equation: $$ \cos{\left( \frac{x}{\pi} \right)}+1=\frac{2}{\phi^6}. $$

Simplifying above, we get another form of the result: $$ I = \pi \arccos{(17-8\sqrt{5})}. $$

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    $\begingroup$ I used RIES also but made some simple adjustments to get a simpler form of the answer $I=8\pi \arcsin(\phi-1)$. $\endgroup$ – Somos Jun 25 '17 at 19:09
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This is not really an answer, but grossly too long for an comment. I didn't know how to simplify it beyond the final solution.

$$I=\int_{-1}^1 \frac{1}{x}\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2x^2+2x+1}{2x^2-2x+1}\right)\text{d}{x}$$

Begin with the substitution of $x=-\cos2a$ $$I=\int_{-1}^1 \frac{1}{-\cos2a}\sqrt{\frac{1-\cos2a}{1+\cos2a}}\ln\left(\frac{2\cos^2 2a-2\cos 2a+1}{2\cos^2 2a-2\cos2a+1}\right)\text{d}{x}$$

By the tangent and cos double angle properties

$$I=\int_{-1}^1 -\sec2a|\tan a|\ln\left(\frac{-2\cos^22a+\cos 4a+2}{2\cos2a+\cos4a+2}\right)\text{d}{a}$$

Were just getting started. Now replace $a=\frac{1}{2}\text{gd}(b)$ where $\text{gd}$ is the Gudermannian function.

$$I=\int_{-1}^1 -\sec(\text{gd}(b))|\tan(\text{gd}(\frac{b}{2}))|\ln\left(\frac{-2\cos^2(\text{gd}(b))+\cos (2\text{gd}(b))+2}{2\cos^2(\text{gd}(b))+\cos (2\text{gd}(b))+2}\right)\text{d}{a}$$

Hehe. Now we get to simplify a bit. This is under the definition of Gudermannian properties.

$$I=\int_{-1}^1 -\text{cosh}\space b|\sinh\frac{b}{2}|\ln\left(\frac{-2\text{sech}^2 b+(\text{sech}^2b+\tanh^2b)+2}{2\text{sech}^2 b+(\text{sech}^2b+\tanh^2b)+2}\right)$$

Now, use properties of $\tanh$ and $\text{sech} $ to simplify even further

$$I=\int_{-1}^1 -\text{cosh}\space b|\sinh\frac{b}{2}|\ln\left(\frac{(1-\text{sech}^2 b)+2}{(1+\text{sech}^2 b)+2}\right)$$

Our goal is to create an $\text{arctanh}$ function, but that will obviously take some serious effort. Factor out a $3$ to generate that $1$ needed even if it makes an ugly factoring.

$$I=\int_{-1}^1 -\text{cosh}\space b|\sinh\frac{b}{2}|\ln\left(\frac{3(1-\frac{\text{sech}^2 b}{3})}{3(1+\frac{\text{sech}^2 b}{3})}\right)$$

And now cut out all of the 3's. After this cut, use a property of $\ln$'s to reciprocate the argument of $\ln$. And multiply 2 and 1/2

$$I=\int_{-1}^1 2\text{cosh}\space b|\sinh\frac{b}{2}|\frac{1}{2}\ln\left(\frac{(1+\frac{\text{sech}^2 b}{3})}{(1-\frac{\text{sech}^2 b}{3})}\right)$$

And what do you know! You're there! Use a property of $\ln$ and $\text{arctanh}$ to generate a much CLEANER form (also by throwing the 2 in front).

$$I=2\int_{-1}^1 \text{cosh}\space b|\sinh\frac{b}{2}|\text{arctanh}(\frac{\text{sech}^2b}{3})$$

This function is even, and we can know that because all parts of what is above, $\cosh b,|\sinh b|, $ etc. all even. So we can do the following.

$$I=4\int_{0}^1 \text{cosh}\space b|\sinh\frac{b}{2}|\text{arctanh}(\frac{\text{sech}^2b}{3})$$

This is just an idea, and like I said not a real solution. I have no idea where to continue beyond this, but I thought it may help to come up with a new idea to solve.

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    $\begingroup$ After further inspection, I messed up my work here. I will leave this post here howver becUse the purpose of the post still holds (ideas to solve) $\endgroup$ – user285523 Nov 22 '15 at 17:42
  • $\begingroup$ Don't you need to change the limits after you make the first change of variable $x = -\cos 2a$? $\endgroup$ – r9m Dec 2 '15 at 16:59
  • $\begingroup$ @user23055 not really. There are lots of mistakes and only consists of only substitution $\endgroup$ – user311151 Apr 2 '16 at 4:48

protected by Ron Gordon Dec 15 '16 at 13:12

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