Optimal. Leaf size=34 \[ \frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac{\sqrt{a^2 x^4+1}}{2 a} \]
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Rubi [A] time = 0.0224494, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6715, 5653, 261} \[ \frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac{\sqrt{a^2 x^4+1}}{2 a} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 5653
Rule 261
Rubi steps
\begin{align*} \int x \sinh ^{-1}\left (a x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sinh ^{-1}(a x) \, dx,x,x^2\right )\\ &=\frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+a^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1+a^2 x^4}}{2 a}+\frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0146358, size = 34, normalized size = 1. \[ \frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac{\sqrt{a^2 x^4+1}}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 31, normalized size = 0.9 \begin{align*}{\frac{1}{2\,a} \left ({x}^{2}a{\it Arcsinh} \left ( a{x}^{2} \right ) -\sqrt{{a}^{2}{x}^{4}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14811, size = 41, normalized size = 1.21 \begin{align*} \frac{a x^{2} \operatorname{arsinh}\left (a x^{2}\right ) - \sqrt{a^{2} x^{4} + 1}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.65478, size = 89, normalized size = 2.62 \begin{align*} \frac{a x^{2} \log \left (a x^{2} + \sqrt{a^{2} x^{4} + 1}\right ) - \sqrt{a^{2} x^{4} + 1}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.251971, size = 27, normalized size = 0.79 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asinh}{\left (a x^{2} \right )}}{2} - \frac{\sqrt{a^{2} x^{4} + 1}}{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 [A] time = 1.29993, size = 54, normalized size = 1.59 \begin{align*} \frac{1}{2} \, x^{2} \log \left (a x^{2} + \sqrt{a^{2} x^{4} + 1}\right ) - \frac{\sqrt{a^{2} x^{4} + 1}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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