Integrand size = 6, antiderivative size = 44 \[ \int x \text {arcsinh}(a x) \, dx=-\frac {x \sqrt {1+a^2 x^2}}{4 a}+\frac {\text {arcsinh}(a x)}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x) \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5776, 327, 221} \[ \int x \text {arcsinh}(a x) \, dx=\frac {\text {arcsinh}(a x)}{4 a^2}-\frac {x \sqrt {a^2 x^2+1}}{4 a}+\frac {1}{2} x^2 \text {arcsinh}(a x) \]
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Rule 221
Rule 327
Rule 5776
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arcsinh}(a x)-\frac {1}{2} a \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {x \sqrt {1+a^2 x^2}}{4 a}+\frac {1}{2} x^2 \text {arcsinh}(a x)+\frac {\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{4 a} \\ & = -\frac {x \sqrt {1+a^2 x^2}}{4 a}+\frac {\text {arcsinh}(a x)}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.91 \[ \int x \text {arcsinh}(a x) \, dx=\frac {-a x \sqrt {1+a^2 x^2}+\left (1+2 a^2 x^2\right ) \text {arcsinh}(a x)}{4 a^2} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )}{2}-\frac {a x \sqrt {a^{2} x^{2}+1}}{4}+\frac {\operatorname {arcsinh}\left (a x \right )}{4}}{a^{2}}\) | \(39\) |
default | \(\frac {\frac {a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )}{2}-\frac {a x \sqrt {a^{2} x^{2}+1}}{4}+\frac {\operatorname {arcsinh}\left (a x \right )}{4}}{a^{2}}\) | \(39\) |
parts | \(\frac {x^{2} \operatorname {arcsinh}\left (a x \right )}{2}-\frac {a \left (\frac {x \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{2}\) | \(65\) |
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int x \text {arcsinh}(a x) \, dx=-\frac {\sqrt {a^{2} x^{2} + 1} a x - {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{4 \, a^{2}} \]
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Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int x \text {arcsinh}(a x) \, dx=\begin {cases} \frac {x^{2} \operatorname {asinh}{\left (a x \right )}}{2} - \frac {x \sqrt {a^{2} x^{2} + 1}}{4 a} + \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int x \text {arcsinh}(a x) \, dx=\frac {1}{2} \, x^{2} \operatorname {arsinh}\left (a x\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.55 \[ \int x \text {arcsinh}(a x) \, dx=\frac {1}{2} \, x^{2} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - \frac {1}{4} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} + \frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{a^{2} {\left | a \right |}}\right )} \]
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Time = 2.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int x \text {arcsinh}(a x) \, dx=x\,\mathrm {asinh}\left (a\,x\right )\,\left (\frac {x}{2}+\frac {1}{4\,a^2\,x}\right )-\frac {x\,\sqrt {a^2\,x^2+1}}{4\,a} \]
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