Apparent paradox in integrating a complex function along closed loop. Let me state the problem right away. I have a complex function which I like to integrate along a circle of radius $r$ centered at origin.
$$
I = \oint_C \frac{dz}{z^2+z}
$$
Here $C$ is circle of radius $r$.
Of course this is very simple to do by finding residue at different poles of the function. And and in fact the value of integral is $2\pi \iota$ when $r<1$ and the value of integral is $0$ when $r>1$.
However if I were to do this by literal replacement of $z$ with $re^{\iota \theta}$ with $\theta \in [0, 2\pi)$. Then we get the integration as,
$$
I = \left[\ln\left(\frac{e^{\iota\theta}}{re^{\iota\theta}+1}\right)\right]_{\theta=0}^{\theta=2\pi}=0
$$
By comparison with the results of residue theorem, we get that the preceding result should have value $2\pi\iota$ if $r<1$ and $0$ if $r>1$. Now if this was not baffling enough, consider this.
The integration result can be re-written as,
$$
I = \left[\iota\theta - \ln(re^{\iota\theta}+1)\right]_{\theta=0}^{\theta=2\pi}=2\pi\iota
$$
I know, I am making some mistake(s) somewhere. May be it has something to do with the multivalued nature of logarithms. But I can not see where I am mistaken. Could someone please point this out to me.

Edit: Calculation of integral $I$ (at the request of @Joppy)
Let's start with the integral,
$$
I = \oint_C \frac{dz}{z^2+z}
$$
Now let's replace $z=re^{\iota\theta}$ and $dz=ire^{\iota\theta}d\theta$. After simplification, this gives,
$$
I = \iota\int_{\theta=0}^{2\pi} \frac{d\theta}{re^{\iota\theta}+1}
$$
Just divide with $e^{\iota\theta}$ in numerator and denominator. This results in,
$$
I = \iota\int_{\theta=0}^{2\pi} \frac{e^{-\iota\theta}d\theta}{r+e^{-\iota\theta}}
$$
This can re-written as,
$$
I = -\int_{\theta=0}^{2\pi} \frac{d(r+e^{-\iota\theta})}{r+e^{-\iota\theta}}
$$
This implies the integral we want is,
$$
I = -\left[\ln(r+e^{-\iota\theta})\right]_{\theta=0}^{2\pi}
=\left[\ln\left(\frac{e^{\iota\theta}}{re^{\iota\theta}+1}\right)\right]_{\theta=0}^{\theta=2\pi}
$$
 A: $\log$ is a priori not a complex function.
As a rule of thumb you can assume that
anything you write that involves complex log is meaningless and wrong, until you learn to be really careful about what a log function is and what they are not.
For example, one thing they are not is being a holomorphic function on $\Bbb C \setminus \{0 \}$, which you assumed all along.
In order to have $[F(\theta)]_{\theta_1}^{\theta_2} = \int_{\theta_1}^{\theta_2}f(\theta) d\theta$ you need first to define a function $F$ on $[\theta_1 ; \theta_2]$ whose derivative is $f(\theta)$.
Whatever function it is that you call $\log$, no matter how you tried to define it, the way you used it to define your $F$ did not give you such a well-behaved $F$. Chances are the $F$ you obtained wasn't even defined on the whole interval, or if it was it was discontinuous somewhere.
If you can plot your F, do so.
Of course since we have no idea what you mean with $\log$ and how you define it, we can't really say more. 
So really, take a closer look at how you have defined a $\log$ function and then take a closer look at how your $F$ really behaves.
