Optimal. Leaf size=25 \[ x \sinh ^{-1}(a x)-\frac{\sqrt{a^2 x^2+1}}{a} \]
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Rubi [A] time = 0.0075172, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5653, 261} \[ x \sinh ^{-1}(a x)-\frac{\sqrt{a^2 x^2+1}}{a} \]
Antiderivative was successfully verified.
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Rule 5653
Rule 261
Rubi steps
\begin{align*} \int \sinh ^{-1}(a x) \, dx &=x \sinh ^{-1}(a x)-a \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{a}+x \sinh ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0072242, size = 25, normalized size = 1. \[ x \sinh ^{-1}(a x)-\frac{\sqrt{a^2 x^2+1}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 26, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax{\it Arcsinh} \left ( ax \right ) -\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 = 1.10788, size = 34, normalized size = 1.36 \begin{align*} \frac{a x \operatorname{arsinh}\left (a x\right ) - \sqrt{a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75271, size = 78, normalized size = 3.12 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \sqrt{a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.152813, size = 20, normalized size = 0.8 \begin{align*} \begin{cases} x \operatorname{asinh}{\left (a x \right )} - \frac{\sqrt{a^{2} x^{2} + 1}}{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 [A] time = 1.35222, size = 47, normalized size = 1.88 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{\sqrt{a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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