Antiderivative of of secx^4

Write (secx)^4 as (secx)^2 * (secx)^2 and then substitute one of the (secx)^2 as 1 + (tanx)^2 and then integrate.

Solution for first integral will be tanx...

Solution of second integral
(secx)^2 * (tanx)^2
Substitute tanx = t.
So (secx)^2dx = dt
So integral reduces to t^2dt..
Solution is (t^3)/3
Replace t with tanx.
So final solution is
tanx + (((tanx)^3)/3) + c

Hint: Write this as (sec²x)(sec²x) and replace one of the sec²x with tan²x + 1