Integrand size = 8, antiderivative size = 34 \[ \int x \text {arcsinh}\left (a x^2\right ) \, dx=-\frac {\sqrt {1+a^2 x^4}}{2 a}+\frac {1}{2} x^2 \text {arcsinh}\left (a x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6847, 5772, 267} \[ \int x \text {arcsinh}\left (a x^2\right ) \, dx=\frac {1}{2} x^2 \text {arcsinh}\left (a x^2\right )-\frac {\sqrt {a^2 x^4+1}}{2 a} \]
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Rule 267
Rule 5772
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \text {arcsinh}(a x) \, dx,x,x^2\right ) \\ & = \frac {1}{2} x^2 \text {arcsinh}\left (a x^2\right )-\frac {1}{2} a \text {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 \text {arcsinh}\left (a x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int x \text {arcsinh}\left (a x^2\right ) \, dx=-\frac {\sqrt {1+a^2 x^4}}{2 a}+\frac {1}{2} x^2 \text {arcsinh}\left (a x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85
method | result | size |
parts | \(\frac {x^{2} \operatorname {arcsinh}\left (a \,x^{2}\right )}{2}-\frac {\sqrt {a^{2} x^{4}+1}}{2 a}\) | \(29\) |
derivativedivides | \(\frac {a \,x^{2} \operatorname {arcsinh}\left (a \,x^{2}\right )-\sqrt {a^{2} x^{4}+1}}{2 a}\) | \(31\) |
default | \(\frac {a \,x^{2} \operatorname {arcsinh}\left (a \,x^{2}\right )-\sqrt {a^{2} x^{4}+1}}{2 a}\) | \(31\) |
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Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int x \text {arcsinh}\left (a x^2\right ) \, dx=\frac {a x^{2} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - \sqrt {a^{2} x^{4} + 1}}{2 \, a} \]
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Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int x \text {arcsinh}\left (a x^2\right ) \, dx=\begin {cases} \frac {x^{2} \operatorname {asinh}{\left (a x^{2} \right )}}{2} - \frac {\sqrt {a^{2} x^{4} + 1}}{2 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int x \text {arcsinh}\left (a x^2\right ) \, dx=\frac {a x^{2} \operatorname {arsinh}\left (a x^{2}\right ) - \sqrt {a^{2} x^{4} + 1}}{2 \, a} \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18 \[ \int x \text {arcsinh}\left (a x^2\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - \frac {\sqrt {a^{2} x^{4} + 1}}{2 \, a} \]
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Time = 2.73 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int x \text {arcsinh}\left (a x^2\right ) \, dx=\frac {x^2\,\mathrm {asinh}\left (a\,x^2\right )}{2}-\frac {\sqrt {a^2\,x^4+1}}{2\,a} \]
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