Integrand size = 10, antiderivative size = 195 \[ \int x^4 \text {arcsinh}(a x)^3 \, dx=-\frac {298 \sqrt {1+a^2 x^2}}{375 a^5}+\frac {76 \left (1+a^2 x^2\right )^{3/2}}{1125 a^5}-\frac {6 \left (1+a^2 x^2\right )^{5/2}}{625 a^5}+\frac {16 x \text {arcsinh}(a x)}{25 a^4}-\frac {8 x^3 \text {arcsinh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arcsinh}(a x)-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^3 \]
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Time = 0.27 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5776, 5812, 5798, 5772, 267, 272, 45} \[ \int x^4 \text {arcsinh}(a x)^3 \, dx=\frac {16 x \text {arcsinh}(a x)}{25 a^4}-\frac {8 x^3 \text {arcsinh}(a x)}{75 a^2}-\frac {3 x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{25 a}-\frac {8 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{25 a^5}-\frac {6 \left (a^2 x^2+1\right )^{5/2}}{625 a^5}+\frac {76 \left (a^2 x^2+1\right )^{3/2}}{1125 a^5}-\frac {298 \sqrt {a^2 x^2+1}}{375 a^5}+\frac {4 x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{25 a^3}+\frac {1}{5} x^5 \text {arcsinh}(a x)^3+\frac {6}{125} x^5 \text {arcsinh}(a x) \]
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Rule 45
Rule 267
Rule 272
Rule 5772
Rule 5776
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {arcsinh}(a x)^3-\frac {1}{5} (3 a) \int \frac {x^5 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^3+\frac {6}{25} \int x^4 \text {arcsinh}(a x) \, dx+\frac {12 \int \frac {x^3 \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{25 a} \\ & = \frac {6}{125} x^5 \text {arcsinh}(a x)+\frac {4 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^3-\frac {8 \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{25 a^3}-\frac {8 \int x^2 \text {arcsinh}(a x) \, dx}{25 a^2}-\frac {1}{125} (6 a) \int \frac {x^5}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {8 x^3 \text {arcsinh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arcsinh}(a x)-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^3+\frac {16 \int \text {arcsinh}(a x) \, dx}{25 a^4}+\frac {8 \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx}{75 a}-\frac {1}{125} (3 a) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = \frac {16 x \text {arcsinh}(a x)}{25 a^4}-\frac {8 x^3 \text {arcsinh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arcsinh}(a x)-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^3-\frac {16 \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx}{25 a^3}+\frac {4 \text {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )}{75 a}-\frac {1}{125} (3 a) \text {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {1+a^2 x}}-\frac {2 \sqrt {1+a^2 x}}{a^4}+\frac {\left (1+a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right ) \\ & = -\frac {86 \sqrt {1+a^2 x^2}}{125 a^5}+\frac {4 \left (1+a^2 x^2\right )^{3/2}}{125 a^5}-\frac {6 \left (1+a^2 x^2\right )^{5/2}}{625 a^5}+\frac {16 x \text {arcsinh}(a x)}{25 a^4}-\frac {8 x^3 \text {arcsinh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arcsinh}(a x)-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^3+\frac {4 \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )}{75 a} \\ & = -\frac {298 \sqrt {1+a^2 x^2}}{375 a^5}+\frac {76 \left (1+a^2 x^2\right )^{3/2}}{1125 a^5}-\frac {6 \left (1+a^2 x^2\right )^{5/2}}{625 a^5}+\frac {16 x \text {arcsinh}(a x)}{25 a^4}-\frac {8 x^3 \text {arcsinh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arcsinh}(a x)-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^3 \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62 \[ \int x^4 \text {arcsinh}(a x)^3 \, dx=\frac {-2 \sqrt {1+a^2 x^2} \left (2072-136 a^2 x^2+27 a^4 x^4\right )+30 a x \left (120-20 a^2 x^2+9 a^4 x^4\right ) \text {arcsinh}(a x)-225 \sqrt {1+a^2 x^2} \left (8-4 a^2 x^2+3 a^4 x^4\right ) \text {arcsinh}(a x)^2+1125 a^5 x^5 \text {arcsinh}(a x)^3}{5625 a^5} \]
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Time = 0.04 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{3}}{5}-\frac {8 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25}-\frac {3 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25}+\frac {4 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25}+\frac {16 a x \,\operatorname {arcsinh}\left (a x \right )}{25}-\frac {4144 \sqrt {a^{2} x^{2}+1}}{5625}+\frac {6 a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )}{125}-\frac {6 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}}{625}+\frac {272 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{5625}-\frac {8 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )}{75}}{a^{5}}\) | \(172\) |
default | \(\frac {\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{3}}{5}-\frac {8 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25}-\frac {3 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25}+\frac {4 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{25}+\frac {16 a x \,\operatorname {arcsinh}\left (a x \right )}{25}-\frac {4144 \sqrt {a^{2} x^{2}+1}}{5625}+\frac {6 a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )}{125}-\frac {6 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}}{625}+\frac {272 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{5625}-\frac {8 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )}{75}}{a^{5}}\) | \(172\) |
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Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.77 \[ \int x^4 \text {arcsinh}(a x)^3 \, dx=\frac {1125 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 225 \, {\left (3 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (9 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 120 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 2 \, {\left (27 \, a^{4} x^{4} - 136 \, a^{2} x^{2} + 2072\right )} \sqrt {a^{2} x^{2} + 1}}{5625 \, a^{5}} \]
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Time = 0.63 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.01 \[ \int x^4 \text {arcsinh}(a x)^3 \, dx=\begin {cases} \frac {x^{5} \operatorname {asinh}^{3}{\left (a x \right )}}{5} + \frac {6 x^{5} \operatorname {asinh}{\left (a x \right )}}{125} - \frac {3 x^{4} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{25 a} - \frac {6 x^{4} \sqrt {a^{2} x^{2} + 1}}{625 a} - \frac {8 x^{3} \operatorname {asinh}{\left (a x \right )}}{75 a^{2}} + \frac {4 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{25 a^{3}} + \frac {272 x^{2} \sqrt {a^{2} x^{2} + 1}}{5625 a^{3}} + \frac {16 x \operatorname {asinh}{\left (a x \right )}}{25 a^{4}} - \frac {8 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{25 a^{5}} - \frac {4144 \sqrt {a^{2} x^{2} + 1}}{5625 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int x^4 \text {arcsinh}(a x)^3 \, dx=\frac {1}{5} \, x^{5} \operatorname {arsinh}\left (a x\right )^{3} - \frac {1}{25} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} + 1} x^{4}}{a^{2}} - \frac {4 \, \sqrt {a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} + 1}}{a^{6}}\right )} a \operatorname {arsinh}\left (a x\right )^{2} - \frac {2}{5625} \, a {\left (\frac {27 \, \sqrt {a^{2} x^{2} + 1} a^{2} x^{4} - 136 \, \sqrt {a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {a^{2} x^{2} + 1}}{a^{2}}}{a^{4}} - \frac {15 \, {\left (9 \, a^{4} x^{5} - 20 \, a^{2} x^{3} + 120 \, x\right )} \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} \]
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Exception generated. \[ \int x^4 \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^4 \text {arcsinh}(a x)^3 \, dx=\int x^4\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]
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