MHB Fractional Logarithmic Integral 02

[alyafey22]

$$\int^1_0 \frac{\log (1+x)}{1+x^2} dx $$

$\log $ is the natural logarithm .

 

[alyafey22]

Here is a hint

Spoiler

Try series expansion and harmonic sums

 

[alyafey22]

Using a result that I proved http://www.mathhelpboards.com/f10/generalized-fractional-logarithm-integral-5467/#post25055

$$\int^t_0 \frac{\log(1+ax)}{1+x}\, dx = - \text{Li}_2 \left( \frac{t}{t+1} \right) +\text{Li}_2 \left(\frac{t-ta}{t+1}\right)-\text{Li}_2(-at)$$

$$\int^{1}_0 \frac{2\log(1+x)}{1+x^2}\, dx = i \text{Li}_2 \left( \frac{1+i}{2} \right)-i \text{Li}_2 \left( \frac{1-i}{2} \right) -i\text{Li}_2 \left(i \right)+i\text{Li}_2 \left(-i \right)$$

Which is numerically equivalent to

$$\int^{1}_0 \frac{2\log(1+x)}{1+x^2}\, dx= \frac{\pi}{4}\log(2)$$

$$\int^{1}_0 \frac{\log(1+x)}{1+x^2}\, dx= \frac{\pi}{8}\log(2)$$

The proof of numerical equivalence is quite long , post it later .