$\begin{array}{rcl}f\left(x\right)& =& \frac{x^9-3x^5+7x^3-x}{{\cos }^{2}x}+{\sec }^{2}x\\ & =& {\sec }^{2}x\left(x^9-3x^5+7x^3-x\right)+{\sec }^{2}x\end{array}$

$\text{ or }\begin{array}{r}{\int }_{-\pi /4}^{\pi /4} f\left(x\right)dx={\int }_{-\pi /4}^{\pi /4} {\sec }^{2}xdx\\ \left[\because {\sec }^{2}x\left(x^9-3x^5+7x^3-x\right)\text{ is an odd function] }\right.\end{array}$

$\begin{array}{r}=2{\int }_0^{\pi /4} {\sec }^{2}xdx\\ ={\left.2\tan x\right|}_0^{\pi /4}=2\end{array}$