Primitive of Inverse Hyperbolic Tangent of x over a over x squared

Theorem

 * $\displaystyle \int \frac {\tanh^{-1} \dfrac x a \ \mathrm d x} {x^2} = \frac {-\tanh^{-1} \dfrac x a} x + \frac 1 {2 a} \ln \left({\frac {x^2} {a^2 - x^2} }\right) + C$

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

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {\sinh^{-1} \frac x a} {x^2}$


 * Primitive of $\dfrac {\cosh^{-1} \frac x a} {x^2}$


 * Primitive of $\dfrac {\coth^{-1} \frac x a} {x^2}$