Primitive of Inverse Hyperbolic Sine of x over a over x

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Theorem

$\displaystyle \int \frac {\sinh^{-1} \dfrac x a \ \mathrm d x} x = \begin{cases} \displaystyle \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^k \left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3} \left({\frac x a}\right)^{2 k + 1} + C & : \left\vert{x}\right\vert < a \\ \displaystyle \frac {\ln^2 \left({\dfrac {2 x} a}\right)} 2 + \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^k \left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3 \left({2 k}\right)^2} \left({\frac a x}\right)^{2 k} + C & : x > a \\ \displaystyle \frac {-\ln^2 \left({\dfrac {2 x} a}\right)} 2 + \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^{k + 1} \left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3 \left({2 k}\right)^2} \left({\frac a x}\right)^{2 k} + C & : x < a \end{cases}$


Proof


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