$\int \frac{1}{x^{2}+6 x+10} d x=$

Step 1 ：This is a problem of integration. The integral is in the form of \(\int \frac{1}{x^{2}+a x+b} dx\).

Step 2 ：To solve this, we can complete the square in the denominator to make it in the form of \(\int \frac{1}{(x+p)^2 + q^2} dx\), which is a standard integral form.

Step 3 ：The integral of this form can be solved as \(\frac{1}{q}\arctan(\frac{x+p}{q}) + C\).

Step 4 ：By completing the square, we find that \(p = -3.0\) and \(q = 1.00000000000000\).

Step 5 ：Substituting these values into the integral, we get \(\arctan(x + 3) + C\).

Step 6 ：Final Answer: \(\boxed{\arctan(x + 3) + C}\)