Primitive of x by Inverse Hyperbolic Secant of x over a

Theorem

 * $\ds \int x \sech^{-1} \frac x a \rd x = \begin {cases}

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


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