A strange derivative-integral discrepancy

[ELB27]

Hi,
I have just encountered a strange (for me) inconsistency with a derivative: If I take the x derivative of $\frac{x}{x+a}$ I get $\frac{x+a-x}{(x+a)^2} = \frac{a}{(x+a)^2}$. However, when I take the integral of the latter I get: $-\frac{a}{x+a} (+constant) ≠ \frac{x}{x+a}$. I have checked the above (simple) calculations with Mathematica to make sure I didn't make any mistake with formulas/arithmetic. Maybe I forget some principle but how is it possible to integrate the derivative of a function and get something else than the function itself back?
Appreciate any help on this.
Just for the record, I have attached a screenshot of Mathematica.

 

[DivergentSpectrum]

very odd.
i used maxima (freeware mathematica):
integrate(a/(a^2+x^2+2*a*x),x)
and got -a/(x+a) like youre saying

then i tried realcalc (a phone app with a symbolic integration feature) and i got x/(a+x)
obviously realcalc is right, i don't understand why a phone app would beat 2 computer algebra systems though. !

 

[MarneMath]

Let the constant of integration be 1. Then it's exactly the same.

 

[ELB27]

Let the constant of integration be 1. Then it's exactly the same.

Ah that's it - thanks. Interesting though how tricky these things can be.

very odd.
i used maxima (freeware mathematica):
integrate(a/(a^2+x^2+2*a*x),x)
and got -a/(x+a) like youre saying

then i tried realcalc (a phone app with a symbolic integration feature) and i got x/(a+x)
obviously realcalc is right, i don't understand why a phone app would beat 2 computer algebra systems though. !

Maybe the app is programmed so that the integral of a derivative of anything would give the same thing back. As for the computer programs - they obviously neglected the constant of integration and took it to be 0.
I wonder if all this operations of common denominator are legal given how they sneakily change the appearance of the function...