Optimal. Leaf size=33 \[ -\frac{a \sqrt{a^2 x^2+1}}{2 x}-\frac{\sinh ^{-1}(a x)}{2 x^2} \]
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Rubi [A] time = 0.0136569, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5661, 264} \[ -\frac{a \sqrt{a^2 x^2+1}}{2 x}-\frac{\sinh ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 264
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)}{x^3} \, dx &=-\frac{\sinh ^{-1}(a x)}{2 x^2}+\frac{1}{2} a \int \frac{1}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2}}{2 x}-\frac{\sinh ^{-1}(a x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0063373, size = 28, normalized size = 0.85 \[ -\frac{a x \sqrt{a^2 x^2+1}+\sinh ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 1.1 \begin{align*}{a}^{2} \left ( -{\frac{{\it Arcsinh} \left ( ax \right ) }{2\,{a}^{2}{x}^{2}}}-{\frac{1}{2\,ax}\sqrt{{a}^{2}{x}^{2}+1}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18445, size = 36, normalized size = 1.09 \begin{align*} -\frac{\sqrt{a^{2} x^{2} + 1} a}{2 \, x} - \frac{\operatorname{arsinh}\left (a x\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94808, size = 88, normalized size = 2.67 \begin{align*} -\frac{\sqrt{a^{2} x^{2} + 1} a x + \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (a x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42686, size = 68, normalized size = 2.06 \begin{align*} \frac{a{\left | a \right |}}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1} - \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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