Optimal. Leaf size=38 \[ \frac {1}{8} \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (3 x^6+2\right )}{24 x^{12}} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 51, 63, 203} \begin {gather*} \frac {\sqrt {x^6-1}}{8 x^6}+\frac {1}{8} \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 51
Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^{13} \sqrt {-1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {1}{8} \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}}{8 x^6}+\frac {1}{16} \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}}{8 x^6}+\frac {1}{8} \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}}{8 x^6}+\frac {1}{8} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 28, normalized size = 0.74 \begin {gather*} \frac {1}{3} \sqrt {x^6-1} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 38, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (2+3 x^6\right )}{24 x^{12}}+\frac {1}{8} \tan ^{-1}\left (\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 34, normalized size = 0.89 \begin {gather*} \frac {3 \, x^{12} \arctan \left (\sqrt {x^{6} - 1}\right ) + {\left (3 \, x^{6} + 2\right )} \sqrt {x^{6} - 1}}{24 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 35, normalized size = 0.92 \begin {gather*} \frac {3 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{6} - 1}}{24 \, x^{12}} + \frac {1}{8} \, \arctan \left (\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 = 32, normalized size = 0.84
method | result | size |
risch | \(\frac {3 x^{12}-x^{6}-2}{24 x^{12} \sqrt {x^{6}-1}}-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{8}\) | \(32\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (3 x^{6}+2\right )}{24 x^{12}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{8}\) | \(48\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }}{2 x^{12}}-\frac {\sqrt {\pi }}{2 x^{6}}+\frac {3 \left (\frac {7}{6}-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{12}+8 x^{6}+8\right )}{16 x^{12}}-\frac {\sqrt {\pi }\, \left (12 x^{6}+8\right ) \sqrt {-x^{6}+1}}{16 x^{12}}-\frac {3 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }}{4}\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 48, normalized size = 1.26 \begin {gather*} \frac {3 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {1}{8} \, \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 = 0.92 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{8}+\frac {\sqrt {x^6-1}}{8\,x^6}+\frac {\sqrt {x^6-1}}{12\,x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.82, size = 124, normalized size = 3.26 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{8} - \frac {i}{8 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{24 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 )}}{8} + \frac {1}{8 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{24 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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