Integrand size = 23, antiderivative size = 153 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}-\frac {40 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^2}+\frac {2 x \text {arcsinh}(a x)^2}{a^3}-\frac {x^3 \text {arcsinh}(a x)^2}{3 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^2} \]
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Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5812, 5798, 5772, 8, 5776, 30} \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 x \text {arcsinh}(a x)^2}{a^3}+\frac {40 x}{9 a^3}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{3 a^2}+\frac {2 x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{9 a^2}-\frac {2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{3 a^4}-\frac {40 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{9 a^4}-\frac {x^3 \text {arcsinh}(a x)^2}{3 a}-\frac {2 x^3}{27 a} \]
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Rule 8
Rule 30
Rule 5772
Rule 5776
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^2}-\frac {2 \int \frac {x \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{3 a^2}-\frac {\int x^2 \text {arcsinh}(a x)^2 \, dx}{a} \\ & = -\frac {x^3 \text {arcsinh}(a x)^2}{3 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^2}+\frac {2}{3} \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx+\frac {2 \int \text {arcsinh}(a x)^2 \, dx}{a^3} \\ & = \frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^2}+\frac {2 x \text {arcsinh}(a x)^2}{a^3}-\frac {x^3 \text {arcsinh}(a x)^2}{3 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^2}-\frac {4 \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{9 a^2}-\frac {4 \int \frac {x \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2}-\frac {2 \int x^2 \, dx}{9 a} \\ & = -\frac {2 x^3}{27 a}-\frac {40 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^2}+\frac {2 x \text {arcsinh}(a x)^2}{a^3}-\frac {x^3 \text {arcsinh}(a x)^2}{3 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^2}+\frac {4 \int 1 \, dx}{9 a^3}+\frac {4 \int 1 \, dx}{a^3} \\ & = \frac {40 x}{9 a^3}-\frac {2 x^3}{27 a}-\frac {40 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^4}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^2}+\frac {2 x \text {arcsinh}(a x)^2}{a^3}-\frac {x^3 \text {arcsinh}(a x)^2}{3 a}-\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^4}+\frac {x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{3 a^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {-2 a x \left (-60+a^2 x^2\right )+6 \left (-20+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)-9 a x \left (-6+a^2 x^2\right ) \text {arcsinh}(a x)^2+9 \left (-2+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{27 a^4} \]
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Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {9 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{3}-9 \operatorname {arcsinh}\left (a x \right )^{3} a^{2} x^{2}-9 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+6 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )-114 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )-2 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}-18 \operatorname {arcsinh}\left (a x \right )^{3}+54 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x -120 \,\operatorname {arcsinh}\left (a x \right )+120 a x \sqrt {a^{2} x^{2}+1}}{27 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(164\) |
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Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {2 \, a^{3} x^{3} - 9 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 9 \, {\left (a^{3} x^{3} - 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 20\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 120 \, a x}{27 \, a^{4}} \]
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Time = 0.52 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {x^{3} \operatorname {asinh}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{3}}{27 a} + \frac {x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{3 a^{2}} + \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{9 a^{2}} + \frac {2 x \operatorname {asinh}^{2}{\left (a x \right )}}{a^{3}} + \frac {40 x}{9 a^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{3 a^{4}} - \frac {40 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{9 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (\sqrt {a^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}\right )} \operatorname {arsinh}\left (a x\right )}{a^{3}} - \frac {a^{2} x^{3} - 60 \, x}{a^{4}}\right )} - \frac {{\left (a^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (a x\right )^{2}}{3 \, a^{3}} \]
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Exception generated. \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {asinh}\left (a\,x\right )}^3}{\sqrt {a^2\,x^2+1}} \,d x \]
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