Primitive of x by Inverse Hyperbolic Secant of x over a

Theorem

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

\dfrac {x^2} 2 \operatorname{sech}^{-1} \dfrac x a - \dfrac {a \sqrt {a^2 - x^2} } 2 + C & : \operatorname{sech}^{-1} \dfrac x a > 0 \\ \dfrac {x^2} 2 \operatorname{sech}^{-1} \dfrac x a + \dfrac {a \sqrt {a^2 - x^2} } 2 + 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 $x \sinh^{-1} \dfrac x a$


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


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


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


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