$\displaystyle \int \text{sech}^2(x) \, dx = \tanh(x) + \bold{C}$
note that an indefinite integral is a "family" of functions that differ by a constant.
$\displaystyle \frac{-2}{e^{2x}+1}$ and $\displaystyle 1 - \frac{2}{e^{2x}+1}$ differ by a constant ... the derivative of either expression equals $\text{sech}^2(x)$
Two captains of industry, Arthur and George, were in a restaurant discussing the state of educational standards, particularly in the field of mathematics. Arthur was convinced they were slipping badly, and that your average college student was completely mathematically illiterate. George, on the other hand, was confident that any student would at least know the basics of calculus.
"I bet you a hundred bucks," said Arthur, "that if you were to ask a random college student a basic question in calculus, he wouldn't understand the question, let alone furnish you with an answer."
"I'll think about that," said George. "Not sure whether to take you up on your bet or not, but I reckon you'd be wrong."
Arthur slipped off to the mens' room at that point, and while he was gone, George called over the waitress Jody. (He knew that was her name because it was written on a badge pinned to her uniform. This appears to be a custom in certain chain diners.)
"I'd like you to help settle a wager between me and my colleague," he said. "When he comes back, I'm going to call you over, and ask you a question, to which you are to answer: one third x cubed."
"Wuntur dex cue?"
"One third x cubed."
"One thurrd ex cuebd."
"That's it, one third x cubed."
"One third ... x cubed."
"That's it, perfect. There's a good tip in it for you."
Arthur returned. George said, "Yes, I think I will take you up on it. A hundred bucks says our waitress can answer such a question. Hey, Jody! What's the indefinite integral of x squared with respect to x?"
"One third x cubed," replied Jody, dutifully.
"You see?" said George, pocketing Arthur's hundred.
As Jody turned away, she called back over her shoulder, "Plus a constant."
George ruefully took Arthur's hundred back out of his pocket and dropped it onto the table.
Still, the calculator result is incorrect:
The constant of integration would have to be 1 and only 1 for the result to be correct.
The range of tanh(x) is (-1, 1), the range of the calculator's result is (-2,0).
The result is not compatible with the basic hyperbolic functions identities such as tanh(x) = sinh(x)/cosh(x).
The calculator result is irreducible to the basic exponential definition of tanh(x).
This statement tells me that you do not understand the concept of an indefinite integral or a general antiderivative. I recommend you research the definitions for both.The constant of integration would have to be 1 and only 1 for the result to be correct.