Primitive of Inverse Hyperbolic Sine Function

Theorem

 * $\ds \int \arsinh x \rd x = x \arsinh x - \sqrt {x^2 + 1} + C$

Proof
From Primitive of $\arsinh \dfrac x a$:
 * $\ds \int \arsinh \frac x a \rd x = x \arsinh \frac x a - \sqrt {x^2 + a^2} + C$

The result follows by setting $a = 1$.

Also see

 * Primitive of $\arcosh x$
 * Primitive of $\artanh x$
 * Primitive of $\arcoth x$