Evaluate: `int(x^2-1)/((x^2+1)sqrt(x^4+1))dx`

Answer

`(1)/(sqrt(2))"tan"^(-1)(sqrt(x^(2)+(1)/(x^(2))))/(sqrt(2))+C`

Explanation

`I=int(x^(2)-1)/((x^(2)+1)sqrt(x^(4)+1))dx`
`=int(x^(2)(1-1//x^(2)))/(x^(2)(x+1//x)sqrt(x^(2)+1//x^(2)))dx `
` =int((1-1//x^(2))dx)/((x+1//x)sqrt((x+1//x)^(2)-2))`
` " Putting " x+1//x=t, " we have " `
` I=int(dt)/(tsqrt(t^(2)-2)) `
` " Again putting " t^(2)-2=y^(2), 2t dt=2ydy, `
`I=int(ydy)/((y^(2)+2)y)=(1)/(sqrt(2))"tan"^(-1)(y)/(sqrt(2))+C`
`=(1)/(sqrt(2))"tan"^(-1)(sqrt(x^(2)+1//x^(2)))/(sqrt(2))+C`