Evaluate: `int1/(x^4+1)dx`

Explanation

We have `int(dx)/((x^(4)+1))=int((x^(2)+1)-(x^(2)-1))/(2(x^(4)+1))dx`
`=(1)/(2)int((x^(2)+1))/((x^(4)+1))dx-(1)/(2)int((x^(2)-1))/((x^(4)+1))dx`
`=(1)/(2)[int((1+(1)/(x^(2))))/((x^(2)+(1)/(x^(2))))dx-int((1-(1)/(x^(2))))/((x^(2)+(1)/(x^(2))))dx]`
[ dividing num . and denom of each integral by `x^(2)`]
`=(1)/(2)[int((1+(1)/(x^(2))))/([(x-(1)/(x))^(2)+2])dx-int((1-(1)/(x^(2))))/([(x+(1)/(x))^(2)-2])dx]`
`=(1)/(2)[int(dt)/([t^(2)+(sqrt(2))^(2)])-int(du)/([u^(2)-(sqrt(2))^(2)])]`
`["Putting"(x-(1)/(x))=t "in the 1 st integral , and" (x+(1)/(x))= u " in the 2 nd"]`
`=(1)/(2){(1)/(sqrt(2))tan^(-1)((t)/(sqrt(2)))-(1)/(2sqrt(2))log|(u-sqrt(2))/(u+sqrt(2))|}+C`
`=(2)/(2sqrt(2))tan^(-1)((x-(1)/(x))/(sqrt(2)))-(1)/(4sqrt(2))log|(x+(1)/(x)-sqrt(2))/(x+(1)/(x)+sqrt(2))|+C`
`=(1)/(2sqrt(2))tan^(-1)((x^(2)-1)/(sqrt(2)x))-(1)/(4sqrt(2))log|(x^(2)+1-sqrt(2)x)/(x^(2)+1+sqrt(2))|+C`.