Evaluate `int(dx)/((1+x-x^(2)))`.

Explanation

We have `int(dx)/((1+x-x^(2)))=-int(dx)/((x^(2)-x-1))`
`=-int(dx)/({(x^(2)-x+(1)/(4))-(5)/(4)})=-int(dx)/({(x-(1)/(2))^(2)-((sqrt(5))/(2))^(2)})`
`=int(dx)/({(sqrt(5)/(2))^(2)-(x-(1)/(2))^(2)})=int(dx)/({(sqrt(5)/(2))^(2)-u^(2)}) "where" (x-(1)/(2))=u`
`=(1)/(2xx(sqrt(5)/(2)))*log|(sqrt(5)/(2)+u)/(sqrt(5)/(2)-u)|+C`
`=(1)/(sqrt(5))log|(sqrt(5)+2u)/(sqrt(5)-2u)|+C=(1)/(sqrt(5))log|(sqrt(5)+2(x-(1)/(2)))/(sqrt(5)-2(x-(1)/(2)))|+C`
`=(1)/(sqrt(5))log|(sqrt(5)+2x-1)/(sqrt(5)-2x+1)|+C=(1)/(sqrt(5))log|((sqrt(5)-1)+2x)/((sqrt(5)+1)-2x)|+C`.