What is the integral of (arctan(ln(x+3)))/(x+3) ?

The integral of (arctan(ln(x+3)))/(x+3) is ln(x+3)arctan(ln(x+3))-1/2 ln|ln^2(x+3)+1|+C

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integral of (arctan(ln(x+3)))/(x+3)

\int \frac{arctan ( ln ( x + 3 ) )}{x + 3} d x

ln (x + 3) arctan (ln (x + 3)) - \frac{1}{2} ln ln^{2} (x + 3) + 1 + C

Solution steps

\int \frac{arctan ( ln ( x + 3 ) )}{x + 3} d x

Apply u-substitution

: \int \frac{arctan ( ln ( u ) )}{u} d u

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

Apply u-substitution

: \int arctan (v) d v

= \int arctan (v) d v

Apply Integration By Parts

: v arctan (v) - \int \frac{v}{v ^{2} + 1} d v

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

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

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

Substitute back

= ln (x + 3) arctan (ln (x + 3)) - \frac{1}{2} ln ln^{2} (x + 3) + 1

Add a constant to the solution

= ln (x + 3) arctan (ln (x + 3)) - \frac{1}{2} ln ln^{2} (x + 3) + 1 + C

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What is the integral of (arctan(ln(x+3)))/(x+3) ?
The integral of (arctan(ln(x+3)))/(x+3) is ln(x+3)arctan(ln(x+3))-1/2 ln|ln^2(x+3)+1|+C