Optimal. Leaf size=155 \[ -\frac {40 \sqrt {a x-1} \sqrt {a x+1}}{27 a^3}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a^3}+\frac {4 x \cosh ^{-1}(a x)}{3 a^2}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a} \]
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Rubi [A] time = 0.47, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5662, 5759, 5718, 5654, 74, 100, 12} \[ -\frac {40 \sqrt {a x-1} \sqrt {a x+1}}{27 a^3}+\frac {4 x \cosh ^{-1}(a x)}{3 a^2}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 100
Rule 5654
Rule 5662
Rule 5718
Rule 5759
Rubi steps
\begin {align*} \int x^2 \cosh ^{-1}(a x)^3 \, dx &=\frac {1}{3} x^3 \cosh ^{-1}(a x)^3-a \int \frac {x^3 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3+\frac {2}{3} \int x^2 \cosh ^{-1}(a x) \, dx-\frac {2 \int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a}\\ &=\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3+\frac {4 \int \cosh ^{-1}(a x) \, dx}{3 a^2}-\frac {1}{9} (2 a) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {4 x \cosh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3-\frac {2 \int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a}-\frac {4 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a}\\ &=-\frac {4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a^3}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {4 x \cosh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3-\frac {4 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a}\\ &=-\frac {40 \sqrt {-1+a x} \sqrt {1+a x}}{27 a^3}-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {4 x \cosh ^{-1}(a x)}{3 a^2}+\frac {2}{9} x^3 \cosh ^{-1}(a x)-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A] time = 0.10, size = 103, normalized size = 0.66 \[ \frac {9 a^3 x^3 \cosh ^{-1}(a x)^3-2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 x^2+20\right )-9 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)^2+6 a x \left (a^2 x^2+6\right ) \cosh ^{-1}(a x)}{27 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 124, normalized size = 0.80 \[ \frac {9 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 9 \, {\left (a^{2} x^{2} + 2\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{2} x^{2} + 20\right )} \sqrt {a^{2} x^{2} - 1}}{27 \, a^{3}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 128, normalized size = 0.83 \[ \frac {\frac {a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{3}}{3}-\frac {2 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}-\frac {\mathrm {arccosh}\left (a x \right )^{2} a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{3}+\frac {4 a x \,\mathrm {arccosh}\left (a x \right )}{3}-\frac {40 \sqrt {a x -1}\, \sqrt {a x +1}}{27}+\frac {2 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )}{9}-\frac {2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{27}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 116, normalized size = 0.75 \[ \frac {1}{3} \, x^{3} \operatorname {arcosh}\left (a x\right )^{3} - \frac {1}{3} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{2} - \frac {2}{27} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x^{2} + \frac {20 \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}}{a^{2}} - \frac {3 \, {\left (a^{2} x^{3} + 6 \, x\right )} \operatorname {arcosh}\left (a x\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.04, size = 138, normalized size = 0.89 \[ \begin {cases} \frac {x^{3} \operatorname {acosh}^{3}{\left (a x \right )}}{3} + \frac {2 x^{3} \operatorname {acosh}{\left (a x \right )}}{9} - \frac {x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{3 a} - \frac {2 x^{2} \sqrt {a^{2} x^{2} - 1}}{27 a} + \frac {4 x \operatorname {acosh}{\left (a x \right )}}{3 a^{2}} - \frac {2 \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{3 a^{3}} - \frac {40 \sqrt {a^{2} x^{2} - 1}}{27 a^{3}} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} x^{3}}{24} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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