Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation}

I am having trouble to calculate the integral. I tried to use by parts method but it didn't help. Wolfram Alpha gives me $0$ as the answer but I don't know how to get it. I also tried to search the similar question here and I got this: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx,$$ yet it didn't help much. Besides, I don't understand the answers there. Could anyone here please help me to calculate the integral preferably (if possible) with elementary ways (high school methods)? Any help would be greatly appreciated. Thank you.

Solution 1:

Update: Finally, a complete solution. Sorry it took so long.

Split the integral up into 3. \begin{align} I &=-\int^{\sqrt{2}}_1\frac{\log{x}}{x}dx+\int^{\sqrt{2}}_1\frac{\log{((x^2-1)^2+1)}}{x}dx-\int^{\sqrt{2}}_1\frac{\log{((x-1)^2+1)}}{x}dx\\ &=-\frac{1}{8}(\log{2})^2+\frac{1}{2}\int^1_0\frac{\log(1+x^2)}{1+x}dx-\int^{\sqrt{2}-1}_0\frac{\log(1+x^2)}{1+x}dx \end{align} The second integral is rather easy to evaluate. \begin{align} \frac{1}{2}\int^1_0\frac{\log(1+x^2)}{1+x}dx &=\frac{1}{2}\int^1_0\int^1_0\frac{x^2}{(1+x)(1+ax^2)}dx \ da\tag1\\ &=\frac{1}{2}\int^1_0\frac{1}{1+a}\int^1_0\frac{1}{1+x}+\frac{x-1}{1+ax^2}dx \ da\\ &=\frac{1}{2}\int^1_0\frac{\log{2}}{1+a}+\frac{\log(1+a)}{2a(1+a)}-\underbrace{\frac{\arctan(\sqrt{a})}{\sqrt{a}(1+a)}}_{\text{Let} \ y=\arctan{\sqrt{a}}}da\\ &=\frac{1}{2}\left[(\log{2})^2+\frac{1}{2}\underbrace{\int^1_0\frac{\log(1+a)}{a}da}_{-\operatorname{Li}_2(-1)=\frac{\pi^2}{12}}-\frac{1}{2}\underbrace{\int^1_0\frac{\log(1+a)}{1+a}da}_{\frac{1}{2}(\log{2})^2}-\frac{\pi^2}{16}\right]\\ &=\frac{3}{8}(\log{2})^2-\frac{\pi^2}{96} \end{align} The third integral can be evaluated using dilogarithms. \begin{align} \int^{\sqrt{2}-1}_0\frac{\log(1+x^2)}{1+x}dx &=\sum_{r=\pm i}\int^{\sqrt{2}-1}_0\frac{\log(r+x)}{1+x}dx\tag2\\ &=-\sum_{r=\pm i}\int^{\frac{\lambda}{\sqrt{2}}}_{\lambda}\log\left(r-1+\frac{\lambda}{y}\right)\frac{dy}{y}\tag3\\ &=-\sum_{r=\pm i}\int^{\frac{r-1}{\sqrt{2}}}_{r-1}\frac{\log(1+y)}{y}-\frac{1}{y}\log\left(\frac{y}{r-1}\right)dy\tag4\\ &=\frac{1}{4}(\log{2})^2+\sum_{r=\pm i}\mathrm{Li}_2\left(\frac{1-r}{\sqrt{2}}\right)-\mathrm{Li}_2(1-r)\tag5\\ &=\frac{1}{4}(\log{2})^2+\mathrm{Li}_2(e^{i\pi/4})+\mathrm{Li}_2(e^{-i\pi/4})-\mathrm{Li}_2(\sqrt{2}e^{i\pi/4})-\mathrm{Li}_2(\sqrt{2}e^{-i\pi/4})\\ &=\frac{1}{4}(\log{2})^2-\frac{\pi^2}{96}\tag6\\ \end{align} It follows that $$I=-\frac{1}{8}(\log{2})^2+\frac{3}{8}(\log{2})^2-\frac{\pi^2}{96}-\frac{1}{4}(\log{2})^2+\frac{\pi^2}{96}=0$$ Explanation
$(1)$: Differentiate under the integral sign
$(2)$: Factorise $1+x^2$, let $r=\pm i$
$(3)$: Let $\displaystyle y=\frac{\lambda}{1+x}$
$(4)$: Let $\lambda=r-1$
$(5)$: Recognise that $\displaystyle\int\frac{\ln(1+y)}{y}dy=-\mathrm{Li}_2(-y)+C$ and $\displaystyle\int\frac{\ln(ay)}{y}dy=\frac{1}{2}\ln^2(ay)+C$
$(6)$: Use the identities here

Solution 2:

Similar problem (two answers)

$$ I = \int_1^{\sqrt{2}}\frac{1}{x}\log\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)\mathrm{d}x $$

Integrate by parts

$$ I = \left.\log(x)\log\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)\right|_1^{\sqrt{2}}- \int_1^{\sqrt{2}}\log(x)\left(\frac{x^6-4 x^5+8 x^4-10 x^2+8 x-4}{x \left(x^2-2 x+2\right) \left(x^4-2 x^2+2\right)}\right)\mathrm{d}x $$

Partial fractions

$$I = \frac{1}{2} \log (2) \log \left(\frac{2}{4 \sqrt{2}-4}\right) - \int_1^{\sqrt{2}}\log(x)\left(-\frac{2 (x-1)}{x^2-2 x+2}+\frac{4 \left(x^3-x\right)}{x^4-2 x^2+2}-\frac{1}{x}\right)\mathrm{d}x$$

It should be easy enough to show that $\int_1^\sqrt{2} \log(x)/x \,\mathrm{d}x = \left.\tfrac12 \log(x)^2\right|_1^\sqrt{2} = \tfrac18\log(2)^2$.

We are left with (after simplification)

$$I = -\frac{1}{8} \log (2) \left(\log (8)-4 \log \left(1+\sqrt{2}\right)\right) - \int_1^{\sqrt{2}}\log(x)\left(-\frac{2 (x-1)}{x^2-2 x+2}+\frac{4\! \left(x^3-x\right)}{x^4-2 x^2+2}\right)\mathrm{d}x$$

I will now show how to finish the evaluation$\def\I{\mathcal{I}}$

$$I_1 = \int_1^\sqrt{2} \frac{2(x-1)\log(x)}{x^2-2x+2} \mathrm{d}x$$

$$I_2 = \int_1^\sqrt{2} \frac{4x(x^2-1)\log(x)}{x^4-2x^2+2} \mathrm{d}x$$

The technique employ involves a partial fractions into the complex numbers. The first expression has a quadratic in the denominator and the second has a quadratic in $x^2$ in the denominator. It should be clear that it is sufficient to solve the equation in terms of

$$\I_1(a) = \int \frac{\log(x)}{x-a} \mathrm{d}x$$

$$\I_2(b) = \int \frac{x\log(x)}{x^2-b^2} \mathrm{d}x$$

The technique used will be geometric series expansions.

$$\I_1(a) = \int \frac{\log(x)}{1-x/a} \mathrm{d}x = -\frac1a \sum_{n=0}^{\infty}\int\log(x)\frac{x^n}{a^n} \mathrm{d}x$$

Integrate by parts

$$\I_1(a) = -\sum_{n=0}^\infty \frac{a^{-(n+1)} x^{n+1}}{(n+1)^2}-\sum_{n=0}^\infty\frac{a^{-(n+1)} x^{n+1} \log (x)}{(n+1)}$$

The first sum is a series expansion of the polylogarithm function of order $2$. The second is also a series for the polylogarithm but it is of order $1$ so it is easily expressible in terms of the logarithm. In all,

$$\I_1(a) = \operatorname{Li}_2\!\left(\frac{x}{a}\right) + \log (x) \log \!\left(1-\frac{x}{a}\right)$$

A similar procedure shows that

$$\I_2(b) = \frac{1}{4}\! \left(\operatorname{Li}_2\left(\frac{x^2}{b^2}\right)+2\log (x) \log \left(1-\frac{x^2}{b^2}\right)\right)$$

Without adding unnecessary details, we need to find

$$2\I_2(\sqrt{1+i}) + 2\I_2(\sqrt{1-i}) -\I_1(1+i) - \I_1(1-i)$$

and then evaluate at $x = 1$ and $x = \sqrt{2}$. A large part of the evaluation goes to zero at $x = 1$ but we are not so fortunate at $x = \sqrt{2}$. What remains is a product of logarithms and a sum of several dilogarithms. These dilogarithms are very easy to evaluate because they all fit under certain dilogarithm identities which can be found here.

For the time being, I will not explicitly evaluate the final sum but that may come in the near future. With the dilogaithm identities, evaluation is more of a nuisance than anything. I will assure you that you do come across the value which zeros out the integral. Feel free to ask if you have any questions.