\int\frac{ax+b}{(x^2+1)^k}\,dx=\frac a2\int\frac{2x}{x^2+1}\,dx+b\int\frac{dx}{(x^2+1)^k}=\frac a2\ln(x^2+1)+b\int\frac{dx}{(x^2+1)^k}