Evaluate: $\int_{-1}^{1}\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}dt$

Question

The value of $$\int_{-1}^{1}\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}dt$$ (where $0<a<1$)

is equal to

(A) $\frac{1}{2a}\left(\ln(\frac{1-a}{1+a})\right)^2$

(B) $\frac{1}{2a}\ln(\frac{1+a}{1-a})$

(C) $\frac{1}{a}\ln(\frac{1+a}{1-a})$

(D) $\frac{1}{2a}\left(\ln(\frac{1+a}{1-a})\right)^2$

(E) $\frac{1}{a}\left(\ln(\frac{1-a}{1+a})\right)^2$

My Attempt

$$I=\int_{-1}^{1}\frac{\ln(1+t)}{1-at}dt-\int_{-1}^{1}\frac{\ln(1-t)}{1-at}dt=\int_{-1}^{1}\frac{\ln(1+t)}{1-at}dt-\int_{-1}^{1}\frac{\ln(1+t)}{1+at}dt$$

$$I=\int_{-1}^{1}\ln(1+t)\left(\frac{1}{1-at}-\frac{1}{1+at}\right)dt=\int_{-1}^{1}\ln(1+t)\left(\frac{2at}{1-a^2t^2}\right)dt$$

After this I am not able to do. Is this approach correct I wonder

Answer 1

\begin{align} &\int_{-1}^1\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}\ \overset{t\to \frac{1-t}{1+t}}{dt}\\ =& -\frac2{1+a}\int_0^\infty \frac{\ln t}{(t+1)(t+ \frac{1-a}{1+a})} \overset{t\to \frac{1-a}{1+a}\frac1t}{dt}\\ =& -\frac2{1+a}\int_0^\infty \frac{\ln \frac{1-a}{1+a}-\ln t}{(t+1)(t+ \frac{1-a}{1+a})} {dt}\\ =& -\frac{\ln \frac{1-a}{1+a}}{1+a}\int_0^\infty \frac{1}{(t+1)(t+ \frac{1-a}{1+a})} {dt}\\ =& -\frac{\ln \frac{1-a}{1+a}}{1+a} \bigg( -\frac{1+a}{2a}{\ln\frac{1-a}{1+a}}\bigg)= \frac{1}{{2a}} \ln^2\frac{1-a}{1+a} \end{align}

Answer 2

$$ \begin{aligned} \int_{-1}^1\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}\mathrm dt&=\int_{-1}^1\ln(1+t)\frac{2at}{1-a^2t^2}\mathrm dt\\ &=-2at\int_{-1}^1\sum_{k\geq 1}\frac{(-1)^kt^{k}}{k}\sum_{n\geq 0}a^{2n}t^{2n}\mathrm dt\\ &=-2\sum_{k\geq 1}\sum_{n\geq 0}\frac{(-1)^ka^{2n+1}}{k}\int_{-1}^1t^{2n+k+1}\mathrm dt\\ &=2\sum_{k\geq 0}\sum_{n\geq 0}\frac{(-1)^ka^{2n+1}}{k+1}\int_{-1}^1t^{2n+k+2}\mathrm dt \end{aligned} $$ In view that $$\int_{-1}^1t^{2n+k+2}=\left\{ \begin{aligned} &0,\ 2m+k+2\ \text{odd}\\ &\frac{2}{2m+2k+3},\ 2m+k+2\ \text{even} \end{aligned} \right. ,$$ this means that only the terms with even $k$ will "survive" because $2n+2+k=$even$+k=$even. So we'll transform $k\to 2k$: $$ \begin{aligned} 2\sum_{k\geq 0}\sum_{n\geq 0}\frac{a^{2n+1}}{2k+1}\int_{-1}^1t^{2(n+k+1)}\mathrm dt&=4\sum_{k\geq 0}\sum_{n\geq 0}\frac{a^{2n+1}}{(2k+1)(2k+2n+3)}\\ &= (...)\\ &=2\sum_{i\geq 0}\sum_{j\geq 0}\frac{a^{2i+2j+1}}{(2i+1)(2j+1)}\\ &=\frac{1}{2a}\left(\ln\left(\frac{1+a}{1-a}\right)\right)^2\implies \mathbf{(A)=(D)}\ \text{is the solution} \end{aligned} $$

Though I kinda cheated in this last part, I'd need help on finding the steps in $(...)$ to convert the double summation I got into the other one since I don't really know how to do that...

Answer 3

Letting $x=\frac{1-t}{1+t}$ transforms the integral into $$ \begin{aligned} I & =-2 \int_0^{\infty} \frac{\ln x}{[(1-a)+(1+a) x](1+x)} d x \\ & =-\frac{2}{1+a} \underbrace{ \int_0^{\infty} \frac{\ln x}{\left(x+k\right)(1+x)} d x}_{J} \end{aligned} $$ where $k= \frac{1-a}{1+a}$.

Putting $x\mapsto\frac{k}{x} $ changes $$ \begin{aligned} J & =\int_0^{\infty} \frac{\ln k-\ln x}{\left(1+x\right)(x+k)} d x \\ & =\ln k \int_0^{\infty} \frac{d x}{\left(1+x\right)(x+k)}-J\\&= \frac{ \ln k}{2} \int_0^{\infty} \frac{d x}{\left(1+x\right)(x+k)}\\&=\frac{\ln k}{2} \cdot \frac{1}{k-1} \int_0^{\infty} \left( \frac{1}{1+x}-\frac{1}{x+k}\right) d x\\&= \frac{\ln ^2 k}{2(k-1)}\\&= \frac{\ln ^2\left(\frac{1-a}{1+a}\right)}{\frac{-4a}{1+a}} \end{aligned} $$ Hence $$ \boxed{I=\frac{1}{2 a} \ln ^2\left(\frac{1-a}{1+a}\right)} $$