Primitive of Inverse Hyperbolic Sine of x over a over x

Theorem

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

\ds \sum_{k \mathop \ge 0} \frac {\paren {-1}^k \paren {2 k + 1}!} {2^{2 k} \paren {k!}^2 \paren {2 k + 1}^3} \paren {\frac x a}^{2 k + 1} + C & : \size x < a \\ \ds \frac {\ln^2 \paren {\dfrac {2 x} a} } 2 + \sum_{k \mathop \ge 0} \frac {\paren {-1}^k \paren {2 k + 1}!} {2^{2 k} \paren {k!}^2 \paren {2 k + 1}^3 \paren {2 k}^2} \paren {\frac a x}^{2 k} + C & : x > a \\ \ds \frac {-\ln^2 \paren {\dfrac {2 x} a} } 2 + \sum_{k \mathop \ge 0} \frac {\paren {-1}^{k + 1} \paren {2 k + 1}!} {2^{2 k} \paren {k!}^2 \paren {2 k + 1}^3 \paren {2 k}^2} \paren {\frac a x}^{2 k} + C & : x < a \end {cases}$

Also see

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


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


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


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


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