Integrand size = 10, antiderivative size = 231 \[ \int x^4 \text {arccosh}(a x)^3 \, dx=-\frac {4144 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^5}-\frac {272 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^3}-\frac {6 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{625 a}+\frac {16 x \text {arccosh}(a x)}{25 a^4}+\frac {8 x^3 \text {arccosh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arccosh}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3 \]
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Time = 0.51 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5883, 5939, 5915, 5879, 75, 102, 12} \[ \int x^4 \text {arccosh}(a x)^3 \, dx=-\frac {8 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{25 a^5}-\frac {4144 \sqrt {a x-1} \sqrt {a x+1}}{5625 a^5}+\frac {16 x \text {arccosh}(a x)}{25 a^4}-\frac {4 x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{25 a^3}-\frac {272 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5625 a^3}+\frac {8 x^3 \text {arccosh}(a x)}{75 a^2}+\frac {1}{5} x^5 \text {arccosh}(a x)^3+\frac {6}{125} x^5 \text {arccosh}(a x)-\frac {3 x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{25 a}-\frac {6 x^4 \sqrt {a x-1} \sqrt {a x+1}}{625 a} \]
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Rule 12
Rule 75
Rule 102
Rule 5879
Rule 5883
Rule 5915
Rule 5939
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {1}{5} (3 a) \int \frac {x^5 \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3+\frac {6}{25} \int x^4 \text {arccosh}(a x) \, dx-\frac {12 \int \frac {x^3 \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a} \\ & = \frac {6}{125} x^5 \text {arccosh}(a x)-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {8 \int \frac {x \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a^3}+\frac {8 \int x^2 \text {arccosh}(a x) \, dx}{25 a^2}-\frac {1}{125} (6 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {6 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{625 a}+\frac {8 x^3 \text {arccosh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arccosh}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3+\frac {16 \int \text {arccosh}(a x) \, dx}{25 a^4}-\frac {6 \int \frac {4 x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{625 a}-\frac {8 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{75 a} \\ & = -\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{225 a^3}-\frac {6 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{625 a}+\frac {16 x \text {arccosh}(a x)}{25 a^4}+\frac {8 x^3 \text {arccosh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arccosh}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {8 \int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{225 a^3}-\frac {16 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a^3}-\frac {24 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{625 a} \\ & = -\frac {16 \sqrt {-1+a x} \sqrt {1+a x}}{25 a^5}-\frac {272 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^3}-\frac {6 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{625 a}+\frac {16 x \text {arccosh}(a x)}{25 a^4}+\frac {8 x^3 \text {arccosh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arccosh}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {8 \int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{625 a^3}-\frac {16 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{225 a^3} \\ & = -\frac {32 \sqrt {-1+a x} \sqrt {1+a x}}{45 a^5}-\frac {272 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^3}-\frac {6 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{625 a}+\frac {16 x \text {arccosh}(a x)}{25 a^4}+\frac {8 x^3 \text {arccosh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arccosh}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3-\frac {16 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{625 a^3} \\ & = -\frac {4144 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^5}-\frac {272 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^3}-\frac {6 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{625 a}+\frac {16 x \text {arccosh}(a x)}{25 a^4}+\frac {8 x^3 \text {arccosh}(a x)}{75 a^2}+\frac {6}{125} x^5 \text {arccosh}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^3 \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.56 \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\frac {-2 \sqrt {-1+a x} \sqrt {1+a x} \left (2072+136 a^2 x^2+27 a^4 x^4\right )+30 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \text {arccosh}(a x)-225 \sqrt {-1+a x} \sqrt {1+a x} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \text {arccosh}(a x)^2+1125 a^5 x^5 \text {arccosh}(a x)^3}{5625 a^5} \]
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Time = 0.06 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{3}}{5}-\frac {8 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {3 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {4 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}+\frac {16 a x \,\operatorname {arccosh}\left (a x \right )}{25}-\frac {4144 \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {6 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )}{125}-\frac {6 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}}{625}-\frac {272 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {8 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )}{75}}{a^{5}}\) | \(190\) |
default | \(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{3}}{5}-\frac {8 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {3 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {4 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{25}+\frac {16 a x \,\operatorname {arccosh}\left (a x \right )}{25}-\frac {4144 \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {6 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )}{125}-\frac {6 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}}{625}-\frac {272 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{5625}+\frac {8 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )}{75}}{a^{5}}\) | \(190\) |
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Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65 \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\frac {1125 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 225 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 30 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (27 \, a^{4} x^{4} + 136 \, a^{2} x^{2} + 2072\right )} \sqrt {a^{2} x^{2} - 1}}{5625 \, a^{5}} \]
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\[ \int x^4 \text {arccosh}(a x)^3 \, dx=\int x^{4} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \]
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Time = 0.25 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.71 \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\frac {1}{5} \, x^{5} \operatorname {arcosh}\left (a x\right )^{3} - \frac {1}{25} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} - 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {a^{2} x^{2} - 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} - 1}}{a^{6}}\right )} a \operatorname {arcosh}\left (a x\right )^{2} - \frac {2}{5625} \, a {\left (\frac {27 \, \sqrt {a^{2} x^{2} - 1} a^{2} x^{4} + 136 \, \sqrt {a^{2} x^{2} - 1} x^{2} + \frac {2072 \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}}{a^{4}} - \frac {15 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \operatorname {arcosh}\left (a x\right )}{a^{5}}\right )} \]
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Exception generated. \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^4 \text {arccosh}(a x)^3 \, dx=\int x^4\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]
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