I would like to know how to prove that $$\sum_{r=1}^\infty \frac{(-1)^{r+1}}{(2r-1)^3}=\frac{\pi^3}{32}$$ I would very much prefer answers that avoid non-elementary functions like the gamma or beta functions, as although I am researching them, I am still in high school and not very familiar with them- but if it is necessary, I will not object :)
I know how to prove the Basel problem using $\sin$ expressed as a product of its factors and its Maclaurin series, and other similar series, such as $$\sum_{r=1}^\infty \frac{(-1)^{r+1}}{r^2}=\frac{\pi^2}{12}$$ but I cannot find a similar method to help evaluate this. It'd also be helpful if I could find an elementary function that I could express as a useful series expansion and then integrate between $0$ and $1$ to evaluate the series I am dealing with such as $$\frac{1}{1^2}-\frac{x^2}{3^2}+\frac{x^4}{5^2}-\frac{x^6}{7^2}+\cdots$$ but I cannot find any function that has this series expansion.
Many thanks for your help.
