Primitive of Inverse Hyperbolic Tangent of x over a

Theorem

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

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 $\sinh^{-1} \dfrac x a$


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


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


 * Primitive of $\operatorname{sech}^{-1} \dfrac x a$


 * Primitive of $\operatorname{csch}^{-1} \dfrac x a$