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

Theorem

 * $\displaystyle \int x^m \operatorname{sech}^{-1} \frac x a \ \mathrm d x = \begin{cases}

\displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{sech}^{-1} \frac x a + \frac 1 {m + 1} \int \frac {x^m} {\sqrt {a^2 - x^2} } \ \mathrm d x + C & : \operatorname{sech}^{-1} \dfrac x a > 0 \\ \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{sech}^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^m} {\sqrt {a^2 - x^2} } \ \mathrm d x + C & : \operatorname{sech}^{-1} \dfrac x a < 0 \\ \end{cases}$

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


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


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