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

The integral of (ln(ln(x))ln(x))/x is 1/2 ln^2(x)ln(ln(x))-(ln^2(x))/4+C

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

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

\frac{1}{2} ln^{2} (x) ln (ln (x)) - \frac{ln ^{2} ( x )}{4} + C

Solution steps

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

Apply u-substitution

= \int u ln (u) d u

Apply Integration By Parts

= \frac{1}{2} u^{2} ln (u) - \int \frac{u}{2} d u

\int \frac{u}{2} d u = \frac{u ^{2}}{4}

= \frac{1}{2} u^{2} ln (u) - \frac{u ^{2}}{4}

Substitute back

u = ln (x)

= \frac{1}{2} ln^{2} (x) ln (ln (x)) - \frac{ln ^{2} ( x )}{4}

Add a constant to the solution

= \frac{1}{2} ln^{2} (x) ln (ln (x)) - \frac{ln ^{2} ( x )}{4} + C

What is the integral of (ln(ln(x))ln(x))/x ?
The integral of (ln(ln(x))ln(x))/x is 1/2 ln^2(x)ln(ln(x))-(ln^2(x))/4+C