Primitive of Inverse Hyperbolic Sine of x over a over x

Proof
For $\size x < a$:

For $x > a$:

For $x < a$:

Also see

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


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


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


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


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