Optimal. Leaf size=43 \[ -\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sinh ^{-1}(a x)^2}{2 x^2} \]
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Rubi [A] time = 0.0804655, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5661, 5723, 29} \[ -\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sinh ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 5723
Rule 29
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^2}{x^3} \, dx &=-\frac{\sinh ^{-1}(a x)^2}{2 x^2}+a \int \frac{\sinh ^{-1}(a x)}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac{\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac{1}{x} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{x}-\frac{\sinh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0310021, size = 43, normalized size = 1. \[ -\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sinh ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 67, normalized size = 1.6 \begin{align*} -{a}^{2}{\it Arcsinh} \left ( ax \right ) -{\frac{a{\it Arcsinh} \left ( ax \right ) }{x}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2\,{x}^{2}}}+{a}^{2}\ln \left ( \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1918, size = 53, normalized size = 1.23 \begin{align*} a^{2} \log \left (x\right ) - \frac{\sqrt{a^{2} x^{2} + 1} a \operatorname{arsinh}\left (a x\right )}{x} - \frac{\operatorname{arsinh}\left (a x\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17422, size = 157, normalized size = 3.65 \begin{align*} \frac{2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, \sqrt{a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{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}^{2}{\left (a x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.48154, size = 149, normalized size = 3.47 \begin{align*} -{\left (a{\left (\frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \frac{{\left | a \right |} \log \left ({\left | x \right |}\right )}{a^{2}}\right )}{\left | a \right |} - \frac{2 \,{\left | a \right |} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1}\right )} a - \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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