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

\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

\int \frac{x ^{2}}{e ^{3 x}} d x

x = 7 y^{2}, x = 4 + 6 y^{2}

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

3 x

\int \frac{x - 9}{x ^{2} + 3 x - 10} d x

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