Evaluate: `int((log)_(e x)edot(log)_(e^2)edot(log)_(e^3x)e)/x dx`

Explanation

`I=int(log_(ex)e*log_(e^(2)x)e*log_(e^(3)x)e)/(x)dx`
`=int(1)/(xlog_(e)ex*log_(e)e^(2)x*log_(e)e^(3)x)dx`
` =int(1)/(x(log_(e)e+log_(e)x)(log_(e)e^(2)+log_(e)x)(log_(e)e^(3)+log_(e)x))dx`
`=int(1)/((1+t)(2+t)(3+t))dt, " where " t=log_(e)x`
`=int((1)/(2)*(1)/(1+t)-(1)/(2+t)+(1)/(3+t))dt " " `[Using partial fractions]
`=(1)/(2)log|1+t|-log|2+t|+log|3+t|+C`
`=(1)/(2)log|1+log_(e)x|-log|2+log_(e)x|+log|3+log|3+log_(e)x|+C`