I am a humble engineer in his first year and I've noticed something curious. I'm from Belgium so English is not my home language (Dutch is) and as I went to a technical school, I haven't received much English (so please excuse my English).
So the integral in question is:
$$\int\frac{4x\cdot\tan(x^2 - 3)}{\cos^2(x^2 - 3)}$$
Solving it by using substitution twice ($t=x^2 - 3$ and then $u=\cos(t)$) gives:
$$\sec^2(x^2 - 3) + C$$
But when differentiating $\tan^2(x^2 - 3)$ I get:
$$\frac{4x\cdot\tan(x^2 - 3)}{\cos^2(x^2 - 3)}$$ this being the original function I integrated.
I also recall: $\sec^2(\theta) - 1 = \tan^2(\theta)$
Thus:
$$\sec^2(x^2 - 3) + C_1 - 1\quad\text{ must equal }\quad\tan^2(x^2 - 3) + C_2$$
This fact scares me because it could mean that any secant or tangent can be transformed at will by adding one or subtracting one as long as you recognize that the constant changes. Is this a correct assumption/paradox?
Regards Enjenir
In your equations above, indeed the solutions are equal. If you re-write $C_1 -1 = C_3$, the $-1$ would be absorbed into $C_1$, and the answers would still differ by a constant.