Optimal. Leaf size=13 \[ \frac{\sinh ^{-1}(a x)^2}{2 a} \]
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Rubi [A] time = 0.021915, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {5675} \[ \frac{\sinh ^{-1}(a x)^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 5675
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx &=\frac{\sinh ^{-1}(a x)^2}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0062127, size = 13, normalized size = 1. \[ \frac{\sinh ^{-1}(a x)^2}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08765, size = 15, normalized size = 1.15 \begin{align*} \frac{\operatorname{arsinh}\left (a x\right )^{2}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47326, size = 51, normalized size = 3.92 \begin{align*} \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.409135, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{asinh}^{2}{\left (a x \right )}}{2 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39752, size = 31, normalized size = 2.38 \begin{align*} \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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