Primitive of Power of x by Inverse Hyperbolic Tangent of x over a

Theorem

 * $\displaystyle \int x^m \tanh^{-1} \frac x a \ \mathrm d x = \frac {x^{m + 1} } {m + 1} \tanh^{-1} \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \ \mathrm d x + C$

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


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


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


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


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