How do you solve the integral $\int \left(\frac{1}{\sec^4(x)}+1\right)dx$ with any method?

Question

I am wondering how many ways there are to solve this integral:

$$\int \left(\frac 1 {\sec^4(x)}+1\right) dx$$

I have a solution with one method, but I would like to know how you would solve the integral with the method you prefer.

If it's possible, could you tell me the country you're from, please? I'm conducting an experiment.

Thanks.

Answer 1

hint:$$\frac{1}{\sec(x)^4}+1=1+\cos(x)^4=\frac{1}{8} (4 \cos (2 x)+\cos (4 x)+11)$$