Integrand size = 8, antiderivative size = 59 \[ \int x \text {arcsinh}(a x)^2 \, dx=\frac {x^2}{4}-\frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 a}+\frac {\text {arcsinh}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^2 \]
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Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5776, 5812, 5783, 30} \[ \int x \text {arcsinh}(a x)^2 \, dx=-\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a}+\frac {\text {arcsinh}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^2+\frac {x^2}{4} \]
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Rule 30
Rule 5776
Rule 5783
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 a}+\frac {1}{2} x^2 \text {arcsinh}(a x)^2+\frac {\int x \, dx}{2}+\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 a} \\ & = \frac {x^2}{4}-\frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 a}+\frac {\text {arcsinh}(a x)^2}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^2 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int x \text {arcsinh}(a x)^2 \, dx=\frac {a^2 x^2-2 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\left (1+2 a^2 x^2\right ) \text {arcsinh}(a x)^2}{4 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arcsinh}\left (a x \right )^{2} \left (a^{2} x^{2}+1\right )}{2}-\frac {\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{2}-\frac {\operatorname {arcsinh}\left (a x \right )^{2}}{4}+\frac {a^{2} x^{2}}{4}+\frac {1}{4}}{a^{2}}\) | \(59\) |
default | \(\frac {\frac {\operatorname {arcsinh}\left (a x \right )^{2} \left (a^{2} x^{2}+1\right )}{2}-\frac {\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{2}-\frac {\operatorname {arcsinh}\left (a x \right )^{2}}{4}+\frac {a^{2} x^{2}}{4}+\frac {1}{4}}{a^{2}}\) | \(59\) |
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int x \text {arcsinh}(a x)^2 \, dx=\frac {a^{2} x^{2} - 2 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{4 \, a^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int x \text {arcsinh}(a x)^2 \, dx=\begin {cases} \frac {x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{2} + \frac {x^{2}}{4} - \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{2 a} + \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.37 \[ \int x \text {arcsinh}(a x)^2 \, dx=\frac {1}{2} \, x^{2} \operatorname {arsinh}\left (a x\right )^{2} + \frac {1}{4} \, a^{2} {\left (\frac {x^{2}}{a^{2}} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{a^{4}}\right )} - \frac {1}{2} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )} \operatorname {arsinh}\left (a x\right ) \]
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Exception generated. \[ \int x \text {arcsinh}(a x)^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \text {arcsinh}(a x)^2 \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^2 \,d x \]
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