Evaluate: `int1/(xsqrt(1+x^3))\ dx`

Explanation

Given: `int1/(xsqrt(1+x^3))\ dx`

`I=int1/(xsqrt(1+x^3))\ dx`
Let `1+x^3=t^2 implies 3x^2dx=2tdt`

`I=int (2t)/(3(t^2-1)t)dt`

`I=(2/3)int (dt)/(t^2-1)`

`I=(2/3)int (dt)/((t+1)(t-1))`

`I=(1/3)int ((t+1)-(t-1))/((t+1)(t-1))dt`

`I=(1/3)[int 1/(t-1)dt-int 1/(t+1)dt]`

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