Optimal. Leaf size=35 \[ \frac {1}{6} x^3 \sqrt {x^6-1}-\frac {1}{6} \log \left (\sqrt {x^6-1}+x^3\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {275, 195, 217, 206} \begin {gather*} \frac {1}{6} x^3 \sqrt {x^6-1}-\frac {1}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 275
Rubi steps
\begin {align*} \int x^2 \sqrt {-1+x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \sqrt {-1+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.20 \begin {gather*} \frac {\left (x^6-1\right ) \left (\sin ^{-1}\left (x^3\right )+\sqrt {1-x^6} x^3\right )}{6 \sqrt {-\left (x^6-1\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 35, normalized size = 1.00 \begin {gather*} \frac {1}{6} x^3 \sqrt {-1+x^6}-\frac {1}{6} \log \left (x^3+\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {x^{6} - 1} x^{3} + \frac {1}{6} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 30, normalized size = 0.86 \begin {gather*} \frac {1}{6} \, \sqrt {x^{6} - 1} x^{3} + \frac {1}{6} \, \log \left ({\left | -x^{3} + \sqrt {x^{6} - 1} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 30, normalized size = 0.86
method | result | size |
trager | \(\frac {x^{3} \sqrt {x^{6}-1}}{6}+\frac {\ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{6}\) | \(30\) |
risch | \(\frac {x^{3} \sqrt {x^{6}-1}}{6}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(38\) |
meijerg | \(\frac {i \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{3} \sqrt {-x^{6}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{12 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 58, normalized size = 1.66 \begin {gather*} -\frac {\sqrt {x^{6} - 1}}{6 \, x^{3} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} - \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \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 = -0.03 \begin {gather*} \int x^2\,\sqrt {x^6-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.30, size = 60, normalized size = 1.71 \begin {gather*} \begin {cases} \frac {x^{9}}{6 \sqrt {x^{6} - 1}} - \frac {x^{3}}{6 \sqrt {x^{6} - 1}} - \frac {\operatorname {acosh}{\left (x^{3} \right )}}{6} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {i x^{3} \sqrt {1 - x^{6}}}{6} + \frac {i \operatorname {asin}{\left (x^{3} \right )}}{6} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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