Integrate question

[parisse]

There is no "constant" in the original answer. The constant comes from the fact that you rewrite the expression in terms of other trigonometric functions. It is of course required since the rewriting command does not know that you are rewriting an antiderivative, which is defined up to a constant.
For the prefered trig functions, it's much simpler for a CAS program to reduce the number of possibilities, that's why functions like sec or csc are immediatly translated to cos and sin equivalents. Hyperbolic trig functions are not necessary since you don't escape the real domain if you rewrite them with exp (unlike trig functions).
I hope this is clear enough and that I will not have to explain this again and again:-)