summation - Calculating the infinite sum $1-frac 1 7+frac 1 9 - frac{1}{15} + frac 1 {17}mp ...=frac{1+sqrt{2}}{8}pi$

Prove that

$$1-\dfrac 1 7+\dfrac 1 9 - \dfrac{1}{15} + \dfrac 1 {17}\mp ...=\dfrac{1+\sqrt{2}}{8}\pi$$

My attempt: I tried to break it into two series
$$(1+1/9+1/17+...)-(1/7+1/15+1/23+...)$$
But I don't know how to proceed. Any hints would be appreciated.

Answer

Using the hints by Mohammad Zuhair Khan and Feng Shao, let

$$f(x):=1-\sum_{n=0}^\infty\left(\frac{x^{8n-1}}{8n-1}-\frac{x^{8n+1}}{8n+1}\right).$$

Then if we differentiate term-wise,

$$f'(x)=-\sum_{n=0}^\infty(x^{8n-2}-x^{8n}).$$

Using the geometric sum formula,

$$f'(x)=-\frac{x^6}{1-x^8}+\frac{x^8}{1-x^8}=-\frac{x^6(1-x^2)}{1-x^8}.$$

Finally,

$$f(1)=1-\int_0^1\frac{x^6(1-x^2)}{1-x^8}dx.$$

https://www.wolframalpha.com/input/?i=integrate+x%5E6(1-x%5E2)%2F(1-x%5E8)+from+0+to+1

I see no easy way to solve the integral, other than by decomposition in simple fractions, which is tedious.