What is the integral of arctanh(x) ?

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

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integral of arctanh(x)

\int arctanh (x) d x

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

Solution steps

\int arctanh (x) d x

Apply Integration By Parts

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

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

\int \frac{x}{1 - x ^{2}} d x = - \frac{1}{2} ln 1 - x^{2}

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

Simplify

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

Add a constant to the solution

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

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