Optimal. Leaf size=65 \[ -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 102, 12,
75} \begin {gather*} -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{9 a^3}+\frac {1}{3} x^3 \cosh ^{-1}(a x)-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{9 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 102
Rule 5883
Rubi steps
\begin {align*} \int x^2 \cosh ^{-1}(a x) \, dx &=\frac {1}{3} x^3 \cosh ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)-\frac {\int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{9 a}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)-\frac {2 \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{9 a}\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+\frac {1}{3} x^3 \cosh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (2+a^2 x^2\right )}{9 a^3}+\frac {1}{3} x^3 \cosh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.47, size = 43, normalized size = 0.66
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \mathrm {arccosh}\left (a x \right )}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} x^{2}+2\right )}{9}}{a^{3}}\) | \(43\) |
default | \(\frac {\frac {a^{3} x^{3} \mathrm {arccosh}\left (a x \right )}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (a^{2} x^{2}+2\right )}{9}}{a^{3}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 48, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arcosh}\left (a x\right ) - \frac {1}{9} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1}}{a^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 52, normalized size = 0.80 \begin {gather*} \frac {3 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{2} x^{2} + 2\right )} \sqrt {a^{2} x^{2} - 1}}{9 \, a^{3}} \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 = 0.13, size = 54, normalized size = 0.83 \begin {gather*} \begin {cases} \frac {x^{3} \operatorname {acosh}{\left (a x \right )}}{3} - \frac {x^{2} \sqrt {a^{2} x^{2} - 1}}{9 a} - \frac {2 \sqrt {a^{2} x^{2} - 1}}{9 a^{3}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{3}}{6} & \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 = -0.02 \begin {gather*} \int x^2\,\mathrm {acosh}\left (a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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