Optimal. Leaf size=48 \[ \frac {1}{144} \sqrt {x^6-1} \left (8 x^{15}-2 x^9-3 x^3\right )-\frac {1}{48} \log \left (\sqrt {x^6-1}+x^3\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.40, number of steps used = 6, 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}{18} \sqrt {x^6-1} x^{15}-\frac {1}{72} \sqrt {x^6-1} x^9-\frac {1}{48} \sqrt {x^6-1} x^3-\frac {1}{48} \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^{14} \sqrt {-1+x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^4 \sqrt {-1+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{48} x^3 \sqrt {-1+x^6}-\frac {1}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {1}{48} x^3 \sqrt {-1+x^6}-\frac {1}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=-\frac {1}{48} x^3 \sqrt {-1+x^6}-\frac {1}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{48} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 1.17 \begin {gather*} \frac {\left (x^6-1\right ) \left (3 \sin ^{-1}\left (x^3\right )+\sqrt {1-x^6} \left (8 x^{12}-2 x^6-3\right ) x^3\right )}{144 \sqrt {-\left (x^6-1\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 48, normalized size = 1.00 \begin {gather*} \frac {1}{144} \sqrt {-1+x^6} \left (-3 x^3-2 x^9+8 x^{15}\right )-\frac {1}{48} \log \left (x^3+\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 42, normalized size = 0.88 \begin {gather*} \frac {1}{144} \, {\left (8 \, x^{15} - 2 \, x^{9} - 3 \, x^{3}\right )} \sqrt {x^{6} - 1} + \frac {1}{48} \, \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^{14}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 40, normalized size = 0.83
method | result | size |
trager | \(\frac {x^{3} \left (8 x^{12}-2 x^{6}-3\right ) \sqrt {x^{6}-1}}{144}-\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{48}\) | \(40\) |
risch | \(\frac {x^{3} \left (8 x^{12}-2 x^{6}-3\right ) \sqrt {x^{6}-1}}{144}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{48 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\) | \(50\) |
meijerg | \(\frac {i \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {i \sqrt {\pi }\, x^{3} \left (-40 x^{12}+10 x^{6}+15\right ) \sqrt {-x^{6}+1}}{60}-\frac {i \sqrt {\pi }\, \arcsin \left (x^{3}\right )}{4}\right )}{12 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 109, normalized size = 2.27 \begin {gather*} -\frac {\frac {3 \, \sqrt {x^{6} - 1}}{x^{3}} + \frac {8 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}} - \frac {3 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}}}{x^{15}}}{144 \, {\left (\frac {3 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} - 1\right )}^{3}}{x^{18}} - 1\right )}} - \frac {1}{96} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {1}{96} \, \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^{14}\,\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 = 3.84, size = 136, normalized size = 2.83 \begin {gather*} \begin {cases} \frac {x^{21}}{18 \sqrt {x^{6} - 1}} - \frac {5 x^{15}}{72 \sqrt {x^{6} - 1}} - \frac {x^{9}}{144 \sqrt {x^{6} - 1}} + \frac {x^{3}}{48 \sqrt {x^{6} - 1}} - \frac {\operatorname {acosh}{\left (x^{3} \right )}}{48} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{21}}{18 \sqrt {1 - x^{6}}} + \frac {5 i x^{15}}{72 \sqrt {1 - x^{6}}} + \frac {i x^{9}}{144 \sqrt {1 - x^{6}}} - \frac {i x^{3}}{48 \sqrt {1 - x^{6}}} + \frac {i \operatorname {asin}{\left (x^{3} \right )}}{48} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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