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

Explanation

`int sqrt(((1+x^(2))/(x^(2)-x^(4))))dx=int(1+x^(2))/(x sqrt((1-x^(4))))dx`
`=int(dx)/(x sqrt((1-x^(4))))+int(x dx)/(sqrt((1-x^(4))))`
`=int(x^(3)dx)/(x^(4) sqrt((1-x^(4))))+int(x dx)/( sqrt((1-x^(4))))`
`= -(1)/(2)int(udu)/((1-u^(2))u)+(1)/(2)int(dv)/( sqrt((1-v^(2))))`
(Putting `1-x^(4)=u^(2),4x^(3)dx=2u du` in the first integral and `xT(2)=v,2x dx=dv` in the second integral)
`=(1)/(2)int(du)/(u^(2)-1)+(1)/(2)sin^(-1)v`
`=(1)/(2)*(1)/(2xx1)log|(u-1)/(u+1)|+(1)/(2)sin^(-1)v +c`
`=(1)/(4)log|(sqrt((1-x^(4)))-1)/(sqrt((1-x^(4)))+1)|+(1)/(2)sin^(-1)(x^(2))+c`