\(\dfrac{1}{x-1}-\dfrac{1}{x+1}-\dfrac{2}{x^2+1}-\dfrac{4}{x^4+1}-\dfrac{8}{x^8+1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{x+1-x+1}{x^2-1}-\dfrac{2}{x^2+1}-\dfrac{4}{x^4+1}-\dfrac{8}{x^8+1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{2}{x^2-1}-\dfrac{2}{x^2+1}-\dfrac{4}{x^4+1}-\dfrac{8}{x^8+1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{2\left(x^2+1-x^2+1\right)}{x^4-1}-\dfrac{4}{x^4+1}-\dfrac{8}{x^8+1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{4}{x^4-1}-\dfrac{4}{x^4+1}-\dfrac{8}{x^8+1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{4\left(x^4+1-x^4+1\right)}{x^8-1}-\dfrac{8}{x^8+1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{8}{x^8-1}-\dfrac{8}{x^8+1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{8\left(x^8+1-x^8+1\right)}{x^{16}-1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{16}{x^{16}-1}-\dfrac{16}{x^{16}+1}\)
\(=\dfrac{16\left(x^{16}+1-x^{16}+1\right)}{x^{32}-1}\)
\(=\dfrac{32}{x^{32}-1}\)
