Primitive of Inverse Hyperbolic Tangent Function/Also presented as

Primitive of Inverse Hyperbolic Tangent Function: Also presented as

This result can also be presented as:

$\ds \int \artanh x \rd x = x \artanh x + \ln \sqrt {1 - x^2} + C$

Proof

From Primitive of $\artanh x$:

$\ds \int \artanh x \rd x = x \artanh x + \frac {\map \ln {1 - x^2} } 2 + C$

The result follows by observing:

$\dfrac 1 2 \log x = \log \sqrt x$

from Logarithm of Power.

$\blacksquare$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals