Integrand size = 10, antiderivative size = 163 \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\frac {45 x \sqrt {1+a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {45 \text {arcsinh}(a x)}{256 a^4}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3 \]
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Time = 0.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5776, 5812, 5783, 327, 221} \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}-\frac {45 \text {arcsinh}(a x)}{256 a^4}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}-\frac {3 x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{16 a}-\frac {3 x^3 \sqrt {a^2 x^2+1}}{128 a}+\frac {9 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{32 a^3}+\frac {45 x \sqrt {a^2 x^2+1}}{256 a^3}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3+\frac {3}{32} x^4 \text {arcsinh}(a x) \]
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Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \text {arcsinh}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^4 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3+\frac {3}{8} \int x^3 \text {arcsinh}(a x) \, dx+\frac {9 \int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a} \\ & = \frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3-\frac {9 \int \frac {\text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{32 a^3}-\frac {9 \int x \text {arcsinh}(a x) \, dx}{16 a^2}-\frac {1}{32} (3 a) \int \frac {x^4}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3+\frac {9 \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx}{128 a}+\frac {9 \int \frac {x^2}{\sqrt {1+a^2 x^2}} \, dx}{32 a} \\ & = \frac {45 x \sqrt {1+a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3-\frac {9 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{256 a^3}-\frac {9 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{64 a^3} \\ & = \frac {45 x \sqrt {1+a^2 x^2}}{256 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2}}{128 a}-\frac {45 \text {arcsinh}(a x)}{256 a^4}-\frac {9 x^2 \text {arcsinh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arcsinh}(a x)+\frac {9 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{16 a}-\frac {3 \text {arcsinh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^3 \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.67 \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\frac {3 a x \left (15-2 a^2 x^2\right ) \sqrt {1+a^2 x^2}+3 \left (-15-24 a^2 x^2+8 a^4 x^4\right ) \text {arcsinh}(a x)-24 a x \sqrt {1+a^2 x^2} \left (-3+2 a^2 x^2\right ) \text {arcsinh}(a x)^2+8 \left (-3+8 a^4 x^4\right ) \text {arcsinh}(a x)^3}{256 a^4} \]
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Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{16}+\frac {9 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x}{32}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}}{128}+\frac {45 a x \sqrt {a^{2} x^{2}+1}}{256}+\frac {27 \,\operatorname {arcsinh}\left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}+1\right ) \operatorname {arcsinh}\left (a x \right )}{32}}{a^{4}}\) | \(141\) |
default | \(\frac {\frac {a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{16}+\frac {9 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x}{32}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}}{128}+\frac {45 a x \sqrt {a^{2} x^{2}+1}}{256}+\frac {27 \,\operatorname {arcsinh}\left (a x \right )}{256}-\frac {9 \left (a^{2} x^{2}+1\right ) \operatorname {arcsinh}\left (a x \right )}{32}}{a^{4}}\) | \(141\) |
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Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.87 \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\frac {8 \, {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 24 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 3 \, {\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1}}{256 \, a^{4}} \]
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Time = 0.51 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98 \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\begin {cases} \frac {x^{4} \operatorname {asinh}^{3}{\left (a x \right )}}{4} + \frac {3 x^{4} \operatorname {asinh}{\left (a x \right )}}{32} - \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1}}{128 a} - \frac {9 x^{2} \operatorname {asinh}{\left (a x \right )}}{32 a^{2}} + \frac {9 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{32 a^{3}} + \frac {45 x \sqrt {a^{2} x^{2} + 1}}{256 a^{3}} - \frac {3 \operatorname {asinh}^{3}{\left (a x \right )}}{32 a^{4}} - \frac {45 \operatorname {asinh}{\left (a x \right )}}{256 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\int { x^{3} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
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Exception generated. \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \text {arcsinh}(a x)^3 \, dx=\int x^3\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]
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