Optimal. Leaf size=231 \[ -\frac {4144 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^5}-\frac {272 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5625 a^3}-\frac {6 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{625 a}+\frac {16 x \cosh ^{-1}(a x)}{25 a^4}+\frac {8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac {6}{125} x^5 \cosh ^{-1}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^3 \]
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Rubi [A]
time = 0.49, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5883, 5939,
5915, 5879, 75, 102, 12} \begin {gather*} -\frac {4144 \sqrt {a x-1} \sqrt {a x+1}}{5625 a^5}-\frac {8 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{25 a^5}+\frac {16 x \cosh ^{-1}(a x)}{25 a^4}-\frac {272 x^2 \sqrt {a x-1} \sqrt {a x+1}}{5625 a^3}-\frac {4 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{25 a^3}+\frac {8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^3+\frac {6}{125} x^5 \cosh ^{-1}(a x)-\frac {6 x^4 \sqrt {a x-1} \sqrt {a x+1}}{625 a}-\frac {3 x^4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{25 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 102
Rule 5879
Rule 5883
Rule 5915
Rule 5939
Rubi steps
\begin {align*} \int x^4 \cosh ^{-1}(a x)^3 \, dx &=\frac {1}{5} x^5 \cosh ^{-1}(a x)^3-\frac {1}{5} (3 a) \int \frac {x^5 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^3+\frac {6}{25} \int x^4 \cosh ^{-1}(a x) \, dx-\frac {12 \int \frac {x^3 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a}\\ &=\frac {6}{125} x^5 \cosh ^{-1}(a x)-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^3-\frac {8 \int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a^3}+\frac {8 \int x^2 \cosh ^{-1}(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 \cosh ^{-1}(a x)}{75 a^2}+\frac {6}{125} x^5 \cosh ^{-1}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^3+\frac {16 \int \cosh ^{-1}(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 \cosh ^{-1}(a x)}{25 a^4}+\frac {8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac {6}{125} x^5 \cosh ^{-1}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(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 \cosh ^{-1}(a x)}{25 a^4}+\frac {8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac {6}{125} x^5 \cosh ^{-1}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(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 \cosh ^{-1}(a x)}{25 a^4}+\frac {8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac {6}{125} x^5 \cosh ^{-1}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(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 \cosh ^{-1}(a x)}{25 a^4}+\frac {8 x^3 \cosh ^{-1}(a x)}{75 a^2}+\frac {6}{125} x^5 \cosh ^{-1}(a x)-\frac {8 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^5}-\frac {4 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a^3}-\frac {3 x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{25 a}+\frac {1}{5} x^5 \cosh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 130, normalized size = 0.56 \begin {gather*} \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 ) \cosh ^{-1}(a x)-225 \sqrt {-1+a x} \sqrt {1+a x} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \cosh ^{-1}(a x)^2+1125 a^5 x^5 \cosh ^{-1}(a x)^3}{5625 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{4} \mathrm {arccosh}\left (a x \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 165, normalized size = 0.71 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 151, normalized size = 0.65 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.70, size = 206, normalized size = 0.89 \begin {gather*} \begin {cases} \frac {x^{5} \operatorname {acosh}^{3}{\left (a x \right )}}{5} + \frac {6 x^{5} \operatorname {acosh}{\left (a x \right )}}{125} - \frac {3 x^{4} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{25 a} - \frac {6 x^{4} \sqrt {a^{2} x^{2} - 1}}{625 a} + \frac {8 x^{3} \operatorname {acosh}{\left (a x \right )}}{75 a^{2}} - \frac {4 x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{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 {acosh}{\left (a x \right )}}{25 a^{4}} - \frac {8 \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{25 a^{5}} - \frac {4144 \sqrt {a^{2} x^{2} - 1}}{5625 a^{5}} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} x^{5}}{40} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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