If ${\displaystyle \int \frac{x^7}{{\left(1-x^2\right)}^5}dx=\frac{1}{8}.\frac{x^8}{{\left(1-x^2\right)}^k}+C}$  then $K=$ ___________

$={\displaystyle \int \frac{x^{-3}}{{\left(\frac{1}{x^2}-1\right)}^5}dx}$

$t=\left(\frac{1}{x^2}-1\right)\Rightarrow dt=-2\frac{1}{x^3}dx$

$\Rightarrow -\frac{dt}{2}=x^{-3}dx$

$-\frac{1}{2}{\displaystyle \int \frac{1}{t^5}dt=-\frac{1}{2}\frac{t^{-4}}{-4}=\frac{1}{8}\frac{1}{{\left(\frac{1}{x^2}-1\right)}^4}}$

$=\frac{1}{8}\cdot \frac{x^8}{{\left(1-x^2\right)}^4}+c$

$\boxed{\therefore k=4}$