Evaluate $\int_0^1 \frac1 {x^2+2x+3}\,\mathrm dx$

I don't understand what do you mean by horrible result

By proceeding with your solution or by evaluating indefinite integral and then applying limits we get

$$\int \frac1 {2+(x+1)^2}dx=\frac{1}{\sqrt{2}}\arctan\left(\frac{x+1}{\sqrt{2}}\right)$$

$$\int_0^1 \frac1 {2+(x+1)^2}dx=\frac{1}{\sqrt{2}}\left[\arctan\left(\sqrt2 \right)-\arctan\left(\frac{1}{\sqrt{2}}\right)\right]$$

You can simplify this to $$\frac{1}{\sqrt{2}}\operatorname {arccot}\left(2 \sqrt{2}\right)$$

But it will still be a horrible result