int tan^3xsec^3x dx

integral (tan^3 (x) sec^3 (x)) dx

Recall the identity,
tan^2 (x) = sec^2 (x) - 1
We substitute it here:
= integral (tan(x) * (sec^2 (x) - 1)(sec^3 (x))) dx
Let u = sec (x)
Thus du = sec(x) tan(x) dx
Substituting,
= integral (u^2 (u^2 - 1)) du
= integral (u^4 - u^2) du
= (1/5)*u^5 - (1/3)*u^3 + C
= (1/5)(sec^5 (x)) - (1/3)(sec^3 (x)) + C

Hope this helps~ :3