What is the integral of (ln(x))/(xln(x)ln(ln(x))) ?

The integral of (ln(x))/(xln(x)ln(ln(x))) is \Ei(ln(ln(x)))+C

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integral of (ln(x))/(xln(x)ln(ln(x)))

\int \frac{ln ( x )}{x ln ( x ) ln ( ln ( x ) )} d x

Ei (ln (ln (x))) + C

Solution steps

\int \frac{ln ( x )}{x ln ( x ) ln ( ln ( x ) )} d x

Cancel the common factor:

ln (x)

= \int \frac{1}{x ln ( ln ( x ) )} d x

Apply u-substitution

= \int \frac{1}{ln ( u )} d u

Apply u-substitution

= \int \frac{e ^{v}}{v} d v

Use the nonelementary integral:

\int \frac{e ^{x}}{x} d x = Ei (x)

= Ei (v)

Substitute back

= Ei (ln (ln (x)))

Add a constant to the solution

= Ei (ln (ln (x))) + C

What is the integral of (ln(x))/(xln(x)ln(ln(x))) ?
The integral of (ln(x))/(xln(x)ln(ln(x))) is \Ei(ln(ln(x)))+C