Optimal. Leaf size=58 \[ -\frac{6 \sqrt{a^2 x^2+1}}{a}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3+6 x \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.0808533, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5653, 5717, 261} \[ -\frac{6 \sqrt{a^2 x^2+1}}{a}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3+6 x \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 5653
Rule 5717
Rule 261
Rubi steps
\begin{align*} \int \sinh ^{-1}(a x)^3 \, dx &=x \sinh ^{-1}(a x)^3-(3 a) \int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3+6 \int \sinh ^{-1}(a x) \, dx\\ &=6 x \sinh ^{-1}(a x)-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3-(6 a) \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{6 \sqrt{1+a^2 x^2}}{a}+6 x \sinh ^{-1}(a x)-\frac{3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3\\ \end{align*}
Mathematica [A] time = 0.0160733, size = 58, normalized size = 1. \[ -\frac{6 \sqrt{a^2 x^2+1}}{a}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+x \sinh ^{-1}(a x)^3+6 x \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 55, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax-3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}+6\,ax{\it Arcsinh} \left ( ax \right ) -6\,\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 = 1.3084, size = 77, normalized size = 1.33 \begin{align*} x \operatorname{arsinh}\left (a x\right )^{3} - \frac{3 \, \sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )^{2}}{a} + \frac{6 \,{\left (a x \operatorname{arsinh}\left (a x\right ) - \sqrt{a^{2} x^{2} + 1}\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0196, size = 205, normalized size = 3.53 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} + 6 \, a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 3 \, \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} - 6 \, \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 = 0.517625, size = 54, normalized size = 0.93 \begin{align*} \begin{cases} x \operatorname{asinh}^{3}{\left (a x \right )} + 6 x \operatorname{asinh}{\left (a x \right )} - \frac{3 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{a} - \frac{6 \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 = 1.43003, size = 132, normalized size = 2.28 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - 3 \, a{\left (\frac{\sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{a^{2}} - \frac{2 \,{\left (x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{\sqrt{a^{2} x^{2} + 1}}{a}\right )}}{a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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