Optimal. Leaf size=53 \[ \frac {\sqrt {x^6-1} \left (48 x^{21}-8 x^{15}-10 x^9-15 x^3\right )}{1152}-\frac {5}{384} \log \left (\sqrt {x^6-1}+x^3\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.57, number of steps used = 7, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {275, 279, 321, 217, 206} \begin {gather*} \frac {1}{24} \sqrt {x^6-1} x^{21}-\frac {1}{144} \sqrt {x^6-1} x^{15}-\frac {5}{576} \sqrt {x^6-1} x^9-\frac {5}{384} \sqrt {x^6-1} x^3-\frac {5}{384} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^{20} \sqrt {-1+x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^6 \sqrt {-1+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{144} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {5}{576} x^9 \sqrt {-1+x^6}-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{192} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {5}{384} x^3 \sqrt {-1+x^6}-\frac {5}{576} x^9 \sqrt {-1+x^6}-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{384} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {5}{384} x^3 \sqrt {-1+x^6}-\frac {5}{576} x^9 \sqrt {-1+x^6}-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{384} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=-\frac {5}{384} x^3 \sqrt {-1+x^6}-\frac {5}{576} x^9 \sqrt {-1+x^6}-\frac {1}{144} x^{15} \sqrt {-1+x^6}+\frac {1}{24} x^{21} \sqrt {-1+x^6}-\frac {5}{384} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 1.15 \begin {gather*} \frac {\left (x^6-1\right ) \left (15 \sin ^{-1}\left (x^3\right )+\sqrt {1-x^6} \left (48 x^{18}-8 x^{12}-10 x^6-15\right ) x^3\right )}{1152 \sqrt {-\left (x^6-1\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 53, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6-1} \left (48 x^{21}-8 x^{15}-10 x^9-15 x^3\right )}{1152}-\frac {5}{384} \log \left (\sqrt {x^6-1}+x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 47, normalized size = 0.89 \begin {gather*} \frac {1}{1152} \, {\left (48 \, x^{21} - 8 \, x^{15} - 10 \, x^{9} - 15 \, x^{3}\right )} \sqrt {x^{6} - 1} + \frac {5}{384} \, \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 [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{6} - 1} x^{20}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 55, normalized size = 1.04 \begin {gather*} \frac {x^{3} \left (48 x^{18}-8 x^{12}-10 x^{6}-15\right ) \sqrt {x^{6}-1}}{1152}-\frac {5 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{384 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 134, normalized size = 2.53 \begin {gather*} -\frac {\frac {15 \, \sqrt {x^{6} - 1}}{x^{3}} + \frac {73 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}} - \frac {55 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}}}{x^{15}} + \frac {15 \, {\left (x^{6} - 1\right )}^{\frac {7}{2}}}{x^{21}}}{1152 \, {\left (\frac {4 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {6 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {4 \, {\left (x^{6} - 1\right )}^{3}}{x^{18}} - \frac {{\left (x^{6} - 1\right )}^{4}}{x^{24}} - 1\right )}} - \frac {5}{768} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {5}{768} \, \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 = -0.02 \begin {gather*} \int x^{20}\,\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 = 6.23, size = 175, normalized size = 3.30 \begin {gather*} \begin {cases} \frac {x^{27}}{24 \sqrt {x^{6} - 1}} - \frac {7 x^{21}}{144 \sqrt {x^{6} - 1}} - \frac {x^{15}}{576 \sqrt {x^{6} - 1}} - \frac {5 x^{9}}{1152 \sqrt {x^{6} - 1}} + \frac {5 x^{3}}{384 \sqrt {x^{6} - 1}} - \frac {5 \operatorname {acosh}{\left (x^{3} \right )}}{384} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{27}}{24 \sqrt {1 - x^{6}}} + \frac {7 i x^{21}}{144 \sqrt {1 - x^{6}}} + \frac {i x^{15}}{576 \sqrt {1 - x^{6}}} + \frac {5 i x^{9}}{1152 \sqrt {1 - x^{6}}} - \frac {5 i x^{3}}{384 \sqrt {1 - x^{6}}} + \frac {5 i \operatorname {asin}{\left (x^{3} \right )}}{384} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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