Integrate $\int\frac{x \sec^2 x \times e^{\sec(x^2)}\:dx}{\cos^2(x^2)}$

Question

Integrate $$I=\int\frac{x \sec^2 x \cdot e^{\sec(x^2)}\:dx}{\cos^2(x^2)}$$

I used the substitution $x^2=t$ we get

$$I=\frac{1}{2} \int \frac{e^{\sec t} \cdot \sec^2(\sqrt{t}) \:dt}{\cos^2 t}$$

Any clue now?