Optimal. Leaf size=183 \[ -\frac {45 x \sqrt {-1+a x} \sqrt {1+a x}}{256 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}-\frac {45 \cosh ^{-1}(a x)}{256 a^4}+\frac {9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac {3}{32} x^4 \cosh ^{-1}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac {3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^3 \]
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Rubi [A]
time = 0.42, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5883, 5939,
5893, 92, 54, 102, 12} \begin {gather*} -\frac {3 \cosh ^{-1}(a x)^3}{32 a^4}-\frac {45 \cosh ^{-1}(a x)}{256 a^4}-\frac {45 x \sqrt {a x-1} \sqrt {a x+1}}{256 a^3}-\frac {9 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{32 a^3}+\frac {9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^3+\frac {3}{32} x^4 \cosh ^{-1}(a x)-\frac {3 x^3 \sqrt {a x-1} \sqrt {a x+1}}{128 a}-\frac {3 x^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 5883
Rule 5893
Rule 5939
Rubi steps
\begin {align*} \int x^3 \cosh ^{-1}(a x)^3 \, dx &=\frac {1}{4} x^4 \cosh ^{-1}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^4 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{16 a}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^3+\frac {3}{8} \int x^3 \cosh ^{-1}(a x) \, dx-\frac {9 \int \frac {x^2 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a}\\ &=\frac {3}{32} x^4 \cosh ^{-1}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{16 a}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^3-\frac {9 \int \frac {\cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a^3}+\frac {9 \int x \cosh ^{-1}(a x) \, dx}{16 a^2}-\frac {1}{32} (3 a) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}+\frac {9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac {3}{32} x^4 \cosh ^{-1}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac {3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^3-\frac {3 \int \frac {3 x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{128 a}-\frac {9 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a}\\ &=-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}+\frac {9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac {3}{32} x^4 \cosh ^{-1}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac {3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^3-\frac {9 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 a^3}-\frac {9 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{128 a}\\ &=-\frac {45 x \sqrt {-1+a x} \sqrt {1+a x}}{256 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}-\frac {9 \cosh ^{-1}(a x)}{64 a^4}+\frac {9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac {3}{32} x^4 \cosh ^{-1}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac {3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^3-\frac {9 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{256 a^3}\\ &=-\frac {45 x \sqrt {-1+a x} \sqrt {1+a x}}{256 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}-\frac {45 \cosh ^{-1}(a x)}{256 a^4}+\frac {9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac {3}{32} x^4 \cosh ^{-1}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac {3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 143, normalized size = 0.78 \begin {gather*} \frac {-3 a x \sqrt {-1+a x} \sqrt {1+a x} \left (15+2 a^2 x^2\right )+24 a^2 x^2 \left (3+a^2 x^2\right ) \cosh ^{-1}(a x)-24 a x \sqrt {-1+a x} \sqrt {1+a x} \left (3+2 a^2 x^2\right ) \cosh ^{-1}(a x)^2+8 \left (-3+8 a^4 x^4\right ) \cosh ^{-1}(a x)^3-45 \log \left (a x+\sqrt {-1+a x} \sqrt {1+a x}\right )}{256 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{3} \mathrm {arccosh}\left (a x \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 142, normalized size = 0.78 \begin {gather*} \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}} \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 = 0.47, size = 170, normalized size = 0.93 \begin {gather*} \begin {cases} \frac {x^{4} \operatorname {acosh}^{3}{\left (a x \right )}}{4} + \frac {3 x^{4} \operatorname {acosh}{\left (a x \right )}}{32} - \frac {3 x^{3} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{3} \sqrt {a^{2} x^{2} - 1}}{128 a} + \frac {9 x^{2} \operatorname {acosh}{\left (a x \right )}}{32 a^{2}} - \frac {9 x \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{32 a^{3}} - \frac {45 x \sqrt {a^{2} x^{2} - 1}}{256 a^{3}} - \frac {3 \operatorname {acosh}^{3}{\left (a x \right )}}{32 a^{4}} - \frac {45 \operatorname {acosh}{\left (a x \right )}}{256 a^{4}} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} x^{4}}{32} & \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 = -0.01 \begin {gather*} \int x^3\,{\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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