Primitive of Inverse Hyperbolic Secant of x over a

Theorem

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

x \operatorname{sech}^{-1} \dfrac x a + a \arcsin \dfrac x a + C & : \operatorname{sech}^{-1} \dfrac x a > 0 \\ x \operatorname{sech}^{-1} \dfrac x a - a \arcsin \dfrac x a + 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 $\sinh^{-1} \dfrac x a$


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


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


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


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