Q.Evaluate $\displaystyle\int \frac{x+1}{x^2+2x+2}\,dx$.

Ten named integral patterns -- $\int dx/(x^2\pm a^2)$, $\int dx/\sqrt{x^2\pm a^2}$, $\int dx/\sqrt{a^2-x^2}$, and their generalisations to a full quadratic $ax^2+bx+c$ under a denominator

or a square root (reduced via completing the square), plus the linear-numerator forms $\int (px+q)\,dx/(ax^2+bx+c)$ and $\int(px+q)\,dx/\sqrt{ax^2+bx+c}$ (numerator split into a multiple of …