Primitive of Inverse Hyperbolic Secant of x over a

Theorem

 * $\ds \int \arsech \frac x a \rd x = \begin{cases}

x \arsech \dfrac x a + a \arcsin \dfrac x a + C & : \arsech \dfrac x a > 0 \\ x \arsech \dfrac x a - a \arcsin \dfrac x a + C & : \arsech \dfrac x a < 0 \end{cases}$

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

let:

and let:

Then:

Also see

 * Primitive of $\arsinh \dfrac x a$


 * Primitive of $\arcosh \dfrac x a$


 * Primitive of $\artanh \dfrac x a$


 * Primitive of $\arcoth \dfrac x a$


 * Primitive of $\arcsch \dfrac x a$