Relationship between input and output voltage on a complex inverting integrator circuit
The formula I got (and I verified it using LTspice) is (in the 'complex' s-domain):

$$\frac{\text{V}_\text{out}\left(\text{s}\right)}{\text{V}_\text{in}\left(\text{s}\right)}=-\frac{1}{\text{R}_1}\cdot\frac{\text{R}_2\cdot\frac{1}{\text{sC}_2}}{\text{R}_2+\frac{1}{\text{sC}_2}}=-\frac{1}{\text{R}_1}\cdot\frac{\text{R}_2}{1+\text{sC}_2\text{R}_2}\tag1$$

Now, we can look at a few cases:


When \$\text{V}_\text{in}\left(t\right)=\hat{\text{v}}\$, where \$\hat{\text{v}}\$ is a constant DC-voltage. We get for the output voltage:
$$\text{V}_\text{out}\left(t\right)=\hat{\text{v}}\cdot\frac{\text{R}_2}{\text{R}_1}\cdot\left(\exp\left(-\frac{t}{\text{C}_2\text{R}_2}\right)-1\right)\tag2$$
When \$\text{V}_\text{in}\left(t\right)=\hat{\text{v}}\cdot\cos\left(\omega t\right)\$. We get for the output voltage:
$$\text{V}_\text{out}\left(t\right)=\frac{\text{R}_2}{\text{R}_1}\cdot\frac{\hat{\text{v}}}{1+\left(\text{R}_2\text{C}_2\omega\right)^2}\cdot\left(\exp\left(-\frac{t}{\text{C}_2\text{R}_2}\right)-\cos\left(\omega t\right)-\text{C}_2\text{R}_2\omega\sin\left(\omega t\right)\right)\tag3$$
When \$\text{V}_\text{in}\left(t\right)=\delta\left(t\right)\$. We get for the output voltage:
$$\text{V}_\text{out}\left(t\right)=-\frac{\exp\left(-\frac{t}{\text{C}_2\text{R}_2}\right)}{\text{C}_2\text{R}_1}\tag4$$
When \$\text{V}_\text{in}\left(t\right)=\theta\left(t\right)\$. We get for the output voltage:
$$\text{V}_\text{out}\left(t\right)=\frac{\text{R}_2}{\text{R}_1}\cdot\left(\exp\left(-\frac{t}{\text{C}_2\text{R}_2}\right)-1\right)\tag5$$




Edit:

The 'complex' s-domain is also known by it's name: the Laplace transform. It plays a very important role in solving this kind of problems.

I solved the output voltage in terms of the input voltage, and it gave me:

$$\text{V}_\text{out}\left(t\right)=-\frac{1}{\text{C}_2\text{R}_1}\int_0^t\text{V}_\text{in}\left(\tau\right)\cdot\exp\left(\frac{\tau-t}{\text{C}_2\text{R}_2}\right)\space\text{d}\tau\tag6$$


  So, no it is not a pure integrator.