Integrand size = 21, antiderivative size = 70 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 x}{3 a^3}-\frac {x^3}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^2} \]
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Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5812, 5798, 8, 30} \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 x}{3 a^3}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}-\frac {2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^4}-\frac {x^3}{9 a} \]
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Rule 8
Rule 30
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^2}-\frac {2 \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^2}-\frac {\int x^2 \, dx}{3 a} \\ & = -\frac {x^3}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^2}+\frac {2 \int 1 \, dx}{3 a^3} \\ & = \frac {2 x}{3 a^3}-\frac {x^3}{9 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{3 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {6 a x-a^3 x^3+3 \left (-2+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^4} \]
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Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.17
method | result | size |
default | \(\frac {3 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )-3 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )-a^{3} x^{3} \sqrt {a^{2} x^{2}+1}-6 \,\operatorname {arcsinh}\left (a x \right )+6 a x \sqrt {a^{2} x^{2}+1}}{9 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(82\) |
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Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {a^{3} x^{3} - 3 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 6 \, a x}{9 \, a^{4}} \]
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Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{3}}{9 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{3 a^{2}} + \frac {2 x}{3 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{3 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=-\frac {1}{9} \, a {\left (\frac {x^{3}}{a^{2}} - \frac {6 \, x}{a^{4}}\right )} + \frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right ) \]
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Exception generated. \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^3\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \]
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