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

Theorem

 * $\ds \int x^m \arcsch \frac x a \rd x = \begin{cases}

\ds \frac {x^{m + 1} } {m + 1} \arcsch \frac x a + \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\ \ds \frac {x^{m + 1} } {m + 1} \arcsch \frac x a - \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0 \\ \end{cases}$

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

let:

and let:

Then:

Also see

 * Primitive of $x^m \arsinh \dfrac x a$


 * Primitive of $x^m \arcosh \dfrac x a$


 * Primitive of $x^m \artanh \dfrac x a$


 * Primitive of $x^m \arcoth \dfrac x a$


 * Primitive of $x^m \arsech \dfrac x a$