Optimal. Leaf size=77 \[ -\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{32 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}-\frac {3 \cosh ^{-1}(a x)}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5883, 102, 12,
92, 54} \begin {gather*} -\frac {3 \cosh ^{-1}(a x)}{32 a^4}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1}}{32 a^3}+\frac {1}{4} x^4 \cosh ^{-1}(a x)-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 5883
Rubi steps
\begin {align*} \int x^3 \cosh ^{-1}(a x) \, dx &=\frac {1}{4} x^4 \cosh ^{-1}(a x)-\frac {1}{4} a \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}+\frac {1}{4} x^4 \cosh ^{-1}(a x)-\frac {\int \frac {3 x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a}\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}+\frac {1}{4} x^4 \cosh ^{-1}(a x)-\frac {3 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{32 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}+\frac {1}{4} x^4 \cosh ^{-1}(a x)-\frac {3 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a^3}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{32 a^3}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{16 a}-\frac {3 \cosh ^{-1}(a x)}{32 a^4}+\frac {1}{4} x^4 \cosh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 71, normalized size = 0.92 \begin {gather*} -\frac {a x \sqrt {-1+a x} \sqrt {1+a x} \left (3+2 a^2 x^2\right )-8 a^4 x^4 \cosh ^{-1}(a x)+6 \tanh ^{-1}\left (\sqrt {\frac {-1+a x}{1+a x}}\right )}{32 a^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 1.48, size = 98, normalized size = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} x^{4} \mathrm {arccosh}\left (a x \right )}{4}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+3 a x \sqrt {a^{2} x^{2}-1}+3 \ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{32 \sqrt {a^{2} x^{2}-1}}}{a^{4}}\) | \(98\) |
default | \(\frac {\frac {a^{4} x^{4} \mathrm {arccosh}\left (a x \right )}{4}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+3 a x \sqrt {a^{2} x^{2}-1}+3 \ln \left (a x +\sqrt {a^{2} x^{2}-1}\right )\right )}{32 \sqrt {a^{2} x^{2}-1}}}{a^{4}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 77, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {arcosh}\left (a x\right ) - \frac {1}{32} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} - 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {a^{2} x^{2} - 1} x}{a^{4}} + \frac {3 \, \log \left (2 \, a^{2} x + 2 \, \sqrt {a^{2} x^{2} - 1} a\right )}{a^{5}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 59, normalized size = 0.77 \begin {gather*} \frac {{\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {a^{2} x^{2} - 1}}{32 \, 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 = 0.20, size = 68, normalized size = 0.88 \begin {gather*} \begin {cases} \frac {x^{4} \operatorname {acosh}{\left (a x \right )}}{4} - \frac {x^{3} \sqrt {a^{2} x^{2} - 1}}{16 a} - \frac {3 x \sqrt {a^{2} x^{2} - 1}}{32 a^{3}} - \frac {3 \operatorname {acosh}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{4}}{8} & \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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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