What is the integral of xarctanh(x) ?

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

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integral of xarctanh(x)

\int x arctanh (x) d x

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

Solution steps

\int x arctanh (x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} arctanh (x) - \int \frac{x ^{2}}{2 ( 1 - x ^{2} )} d x

= \frac{1}{2} x^{2} arctanh (x) - \int \frac{x ^{2}}{2 ( 1 - x ^{2} )} d x

\int \frac{x ^{2}}{2 ( 1 - x ^{2} )} d x = \frac{1}{4} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 x)

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

Add a constant to the solution

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

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