Integration question about absolute value

y = sinh^-1(tan(x))
I integrated this as following:
sec^2(x) / (sqrt(tan^2(x) +1)
sec^2(x) / (sqrt(sec^2(x))
sec^2(x) / sec(x)
sec(x) + c
but the book says the final answer is the the absolute value of sec(x) so this was the book's answer: |sec(x)|

Can anyone please explain why there should be an absolute value around the secant. Thank You in advance.

This is a derivative, and I see you corrected yourself in "Additional Details", but then you still went on to say that you "integrated y" instead of "differentiating y".

Anyway, your first step is correct:
dy/dx = 1/√(tan²x + 1) * d/dx (tanx) = sec²x / √(tan²x + 1)
dy/dx = sec²x / √sec²x

Now depending on value of x, sec(x) could be positive or negative.
But √sec²x is always positive.
So when sec(x) > 0, then √sec²x = sec(x)
but when sec(x) < 0, then √sec²x = −sec(x)
Therefore √sec²x = |sec(x)|

dy/dx = sec²x / |sec(x)|
dy/dx = |sec(x)|