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

Theorem

 * $\displaystyle \int x^2 \tanh^{-1} \frac x a \ \mathrm d x = \frac {a x^2} 6 + \frac {x^3} 3 \tanh^{-1} \frac x a + \frac {a^3} 6 \ln \left({a^2 - x^2}\right) + C$

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 $x^2 \sinh^{-1} \dfrac x a$


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


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