Question

$\int \frac{x^2+9}{x^4-2x^2+81}dx$

Answer 1

$\frac{x^2+9}{x^4-2x^2+81}=\frac{1+\frac{9}{x^2}}{x^2+\frac{81}{x^2}-2}=\frac{1+\frac{9}{x^2}}{(x-\frac{9}{x}{)}^2+2\left(9\right)-2}$

Put $t=x-\frac{9}{x}⟹dt=(1+\frac{9}{x^2})dx$

$\int \frac{x^2+9}{x^4-2x^2+81}dx=\int \frac{dt}{t^2+16}=\frac{1}{4}{\text{tan}}^{-1}(\frac{t}{4})+C=\frac{1}{4}{\text{tan}}^{-1}(\frac{x-\frac{9}{x}}{4})+C$