Evaluate `int sqrt(x^2+1)[log(x^2+1)-2logx]/(x^4)dx`

Explanation

Given,
`int (sqrt(x^2+1)[log(x^2+1)−2logx])/(x^4) dx`
`=int (sqrt(x^2+1)log((x^2+1)/(x^2)))/(x^4)dx`
`=int sqrt((x^2+1)/(x^2))log((x^2+1)/(x^2))×1/(x^3)dx`
put `int sqrt((x^2+1)/(x^2))=t, (x^2+1)/(x^2)=t^2`
`−2/(x^3)dx=2t dt`
`1/(x^3)dx=−tdt`
`=int t.logt^2(−t)dt`
`=-int t2×log(t^2)×dt=[logt^2×(t^3)/3−sqrt(1/(t^2) xx 2t xx (t^3)/3]`
`=−[logt^2×(t^3)/3−2/3×(t^3)/3]+C`
`=−1/3 t^3 logt^2+2/3 t^3+C⇒−1/3 t^3[logt^2−2/3]+C`
`=−1/3[(x^2+1)/(x^2)]^(3/2)[log((x^2+1)/(x^2))−2/3]+C`