Evaluate $\int_{1}^{2} \int_{1}^{3} \frac{xy}{x^2 +y^2} \; dx\; dy$ 
How does one evaluate the following? $$\int_{1}^{2} \int_{1}^{3} \frac{xy}{x^2 +y^2} \, dx\, dy$$

It gets very tedious when I use a substitution method. Is there a short way to evaluate it? 

Notice that : $\int \frac{xy}{x^2+y^2}dx = \frac{y}{2}\ln(x^2+y^2) + C $
$$\int_{1}^{2} \int_{1}^{3} \frac{xy}{x^2 +y^2} \; dx\; dy = \int_1^2 \frac{y}{2}\ln(9+y^2)-\frac{y}{2}\ln(1+y^2)dy= \int_1^4 \frac{\ln(9+u)}{4}-\frac{\ln(1+u)}{4}du$$ where $u=y^2$ and $du=2ydy $
Setting $t=9+u$ and $dt=du$ and $s=1+u$ and $ds=du$ we have 
$$\int_{10}^{13} \frac{\ln(t)}{4}dt-\int_2^5 \frac{\ln(s)}{4}ds$$
And then use integration by parts
