Optimal. Leaf size=36 \[ \frac {1}{24} \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (x^6-2\right )}{24 x^{12}} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 47, 51, 63, 203} \begin {gather*} \frac {\sqrt {x^6-1}}{24 x^6}+\frac {1}{24} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\sqrt {x^6-1}}{12 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^6}}{x^{13}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^3} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {\sqrt {-1+x^6}}{24 x^6}+\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {\sqrt {-1+x^6}}{24 x^6}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=-\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {\sqrt {-1+x^6}}{24 x^6}+\frac {1}{24} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 28, normalized size = 0.78 \begin {gather*} \frac {1}{9} \left (x^6-1\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};1-x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 36, normalized size = 1.00 \begin {gather*} \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{24 x^{12}}+\frac {1}{24} \tan ^{-1}\left (\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 31, normalized size = 0.86 \begin {gather*} \frac {x^{12} \arctan \left (\sqrt {x^{6} - 1}\right ) + \sqrt {x^{6} - 1} {\left (x^{6} - 2\right )}}{24 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 63, normalized size = 1.75 \begin {gather*} \frac {1}{96} \, \pi + \frac {\sqrt {x^{6} - 1} - \frac {1}{\sqrt {x^{6} - 1}}}{24 \, {\left ({\left (\sqrt {x^{6} - 1} - \frac {1}{\sqrt {x^{6} - 1}}\right )}^{2} + 4\right )}} + \frac {1}{48} \, \arctan \left (\frac {x^{6} - 2}{2 \, \sqrt {x^{6} - 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 30, normalized size = 0.83
method | result | size |
risch | \(\frac {x^{12}-3 x^{6}+2}{24 x^{12} \sqrt {x^{6}-1}}-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{24}\) | \(30\) |
trager | \(\frac {\left (x^{6}-2\right ) \sqrt {x^{6}-1}}{24 x^{12}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\) | \(46\) |
meijerg | \(-\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }}{x^{12}}-\frac {\sqrt {\pi }}{x^{6}}+\frac {\left (\frac {1}{2}-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (x^{12}-8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right ) \sqrt {-x^{6}+1}}{8 x^{12}}-\frac {\ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }}{2}\right )}{12 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 46, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}} - \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {1}{24} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 35, normalized size = 0.97 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{24}-\frac {\frac {\sqrt {x^6-1}}{24}-\frac {{\left (x^6-1\right )}^{3/2}}{24}}{x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.58, size = 124, normalized size = 3.44 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{24} - \frac {i}{24 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{8 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i}{12 x^{15} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{24} + \frac {1}{24 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{8 x^{9} \sqrt {1 - \frac {1}{x^{6}}}} + \frac {1}{12 x^{15} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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