Yet another difficult logarithmic integral

Solution 1:

Long Comment: Notes on evaluating $I_A$:

I first found the equivalent integral $$I_A=2 \int_0^1 \frac{\sinh ^{-1}(x)}{x \sqrt{1-x^2}} \, dx\tag{1}$$ Integrating (1) by parts I then found $$I_A=2 \int_0^1 \frac{\log \left(\frac{\sqrt{1-x^2}+1}{x}\right)}{\sqrt{x^2+1}} \, dx\tag{2}$$

Then since $$\frac{1}{\sqrt{x^2+1}}=\sum _{n=0}^{\infty } \frac{(-1)^n \binom{2 n}{n}}{4^n}\, x^{2 n}$$

we obtain using (2) $$I_A=2 \sum _{n=0}^{\infty } \frac{(-1)^n \binom{2 n}{n}}{4^n}\, \int_0^1 x^{2 n}\log \left(\frac{\sqrt{1-x^2}+1}{x}\right) \, dx $$

Then using Mathematica

$$\int_0^1 x^{2 n} \log \left(\frac{\sqrt{1-x^2}+1}{x}\right) \, dx=\frac{\sqrt{\pi }\, \Gamma \left(n+\frac{3}{2}\right)}{(2 n+1)^2\, \Gamma (n+1)}$$

$$I_A=2 \sum _{n=0}^{\infty } \frac{(-1)^n \binom{2 n}{n}}{4^n}\,\frac{\sqrt{\pi }\, \Gamma \left(n+\frac{3}{2}\right)}{(2 n+1)^2\, \Gamma (n+1)}=\pi \, _3F_2\left(\frac{1}{2},\frac{1}{2},\frac{1}{2};1,\frac{3}{2};-1\right)$$

UPDATE 23/03/2019

In thinking about Jack D'Aurizio's comment below: I think the relationship between the integrals and Euler sums can perhaps be found by proving and then making use of (3) and (4) $$\int_0^1 x^n \tanh ^{-1}\left(\sqrt{x}\right) \, dx=\frac{1}{n+1}\sum _{k=1}^{n+1} \frac{1}{2 k-1}\tag{3}$$

$$\int_0^1 x^{2n} \tanh ^{-1}\left(\sqrt{x}\right) \, dx=\frac{1}{2n+1}\sum _{k=1}^{2n+1} \frac{1}{2 k-1}\tag{4}$$

for Integrals A and B respectively.

In regard to integral A this looks difficult because it involves simplifying something like:

$$I_A=\frac{1}{\sqrt{2}} \int_0^1 \left(\sum _{n=0}^{\infty } \binom{2 n}{n} \frac{x^n}{4^n}\right) \left(\sum _{n=0}^{\infty } \binom{2 n}{n} \frac{x^n}{8^n} \right) \tanh ^{-1}\left(\sqrt{x}\right) \,dx$$

I get stuck trying to simplify the Cauchy Product.

The proof may be easier in regard to integral B since

$$\frac{1}{\sqrt{1-x^2}}=\sum _{n=0}^{\infty } \binom{2 n}{n} \frac{x^{2 n}} {4^n} $$

giving

$$I_B=\sqrt{2} \int_0^1\frac{ \tanh ^{-1}\left(\sqrt{x}\right)}{\sqrt{(1-x) (x+1)}}\; dx =\sqrt{2}\; \sum _{n=0}^{\infty } \frac{\binom{2 n}{n} \sum _{k=1}^{2 n+1} \frac{1}{2 k-1}}{4^n (2 n+1)}$$