Primitive of x by Inverse Hyperbolic Tangent of x over a

Theorem

 * $\ds \int x \artanh \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \artanh \frac x a + C$

Proof
With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Also see

 * Primitive of $x \arsinh \dfrac x a$


 * Primitive of $x \arcosh \dfrac x a$


 * Primitive of $x \arcoth \dfrac x a$


 * Primitive of $x \arsech \dfrac x a$


 * Primitive of $x \arcsch \dfrac x a$