Optimal. Leaf size=13 \[ -\frac{1}{2 a \sinh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.0354877, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {5675} \[ -\frac{1}{2 a \sinh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 5675
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx &=-\frac{1}{2 a \sinh ^{-1}(a x)^2}\\ \end{align*}
Mathematica [A] time = 0.0073274, size = 13, normalized size = 1. \[ -\frac{1}{2 a \sinh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 12, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.32081, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{2 \, a \operatorname{arsinh}\left (a x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.6282, size = 55, normalized size = 4.23 \begin{align*} -\frac{1}{2 \, a \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.25273, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{2 a \operatorname{asinh}^{2}{\left (a x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26494, size = 31, normalized size = 2.38 \begin{align*} -\frac{1}{2 \, a \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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