Trigonometric Integral Question

Have a tricky one here...?

Integral of sec^3x*tan^2x dx

I can get part way thorough it, but can never get the right answer.
Thanks

For the sake of adopting a reasonably efficient shorthand, let t = tan x and s = sec x.
Let us call our integral f = integral (s^3t^2 dx) = integral (s^3(s^2 - 1) dx) = integral (s^5 - s^3 dx)

Integrating by parts,

u = (s^2)t, dv = st dx, so that
du = s^2(s^2) + t(2s)(st) dx = s^4 + 2s^2t^2 dx = s^4 + 2s^2(s^2 - 1) dx = 3s^4 - 2s^2 dx
v = s

f = integral (u dv) = uv - integral (v du)
f = (s^2t)s - integral (s(3s^4 - 2s^2)) dx
f = s^3t - integral (3s^5 - 2s^3) dx
f = s^3t - integral (3s^5 - 2s^3) dx + integral (s^3 dx) - integral (s^3 dx)
f = s^3t - integral (3s^5 - 3s^3) dx - integral (s^3 dx)
f = s^3t - 3integral (s^5 - s^3) dx - integral (s^3 dx)
f = s^3t - 3f - integral (s^3 dx)
4f = s^3t - integral (s^3 dx)
f = (1/4)s^3t - (1/4)integral (s^3 dx)

So we have reduced finding f to finding g = integral (s^3 dx). Integrating by parts again,
u = s, dv = s^2 dx
du = st dx, v = t

g = integral (u dv) = uv - integral (v du)
g = st - integral (st^2 dx)
g = st - integral (s(s^2 - 1) dx)
g = st - integral (s^3 - s) dx)
g = st - integral (s^3 dx) + integral ( s dx)
g = st - g + integral (s dx)
2g = st + integral (s dx)
g = (1/2)st + (1/2) integral (s dx)

Have faith! All we need to do now is find h = integral (s dx). It is known that h = ln |s + t|.
I will spare you the derivation and refer you to a soothing demonstration of this fact on YouTube (see source link below). It is not intuitive, but is something every Calculus student must grapple with at some point.

So plugging into our chain of expressions
g = (1/2)st + (1/2)h
g = (1/2)st + (1/2)ln|s + t|

f = (1/4)s^3t - (1/4)g
f = (1/4)s^3t - (1/4)((1/2)st + (1/2)ln|s + t|)
f = (1/4)s^3t - (1/8)st - (1/8)ln|s + t|

Writing it all out, and adding in the +C (which may cost you a test point if you forget)

Integral (sec^3x*tan^2x dx)
= (1/4)(sec^3x)(tan x) - (1/8)(sec x)(tan x) - (1/8)ln|(sec x) + (tan x)| + C

Note: As an exercise, you might want to take the derivative and verify that this does in fact differentiate to (sec^3x)(tan^2x). I actually found a sign error that way resulting in my having to edit this post!