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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int arctan (u) d u

Apply Integration By Parts

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

\int \frac{u}{u ^{2} + 1} d u = \frac{1}{2} ln u^{2} + 1

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

Substitute back

u = ln (x)

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

Add a constant to the solution

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

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