Optimal. Leaf size=42 \[ \frac {\left (1-4 x^3\right ) \sqrt {x^6-1}}{6 x^6}-\tan ^{-1}\left (\frac {x^3+1}{\sqrt {x^6-1}}\right ) \]
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Rubi [F] time = 0.68, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7 \left (1+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7 \left (1+x^3\right )} \, dx &=\int \left (-\frac {\sqrt {-1+x^6}}{x^7}+\frac {2 \sqrt {-1+x^6}}{x^4}-\frac {2 \sqrt {-1+x^6}}{x}+\frac {2 \sqrt {-1+x^6}}{3 (1+x)}+\frac {2 (-1+2 x) \sqrt {-1+x^6}}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {2}{3} \int \frac {(-1+2 x) \sqrt {-1+x^6}}{1-x+x^2} \, dx+2 \int \frac {\sqrt {-1+x^6}}{x^4} \, dx-2 \int \frac {\sqrt {-1+x^6}}{x} \, dx-\int \frac {\sqrt {-1+x^6}}{x^7} \, dx\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^6\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {2}{3} \int \left (\frac {2 \sqrt {-1+x^6}}{-1-i \sqrt {3}+2 x}+\frac {2 \sqrt {-1+x^6}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1+i \sqrt {3}+2 x} \, dx\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1+i \sqrt {3}+2 x} \, dx\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {2 \sqrt {-1+x^6}}{3 x^3}+\frac {1}{2} \tan ^{-1}\left (\sqrt {-1+x^6}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1+i \sqrt {3}+2 x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.15, size = 37, normalized size = 0.88 \begin {gather*} \frac {1}{6} \left (3 \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (1-4 x^3\right )}{x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 44, normalized size = 1.05 \begin {gather*} \frac {\left (1-4 x^3\right ) \sqrt {-1+x^6}}{6 x^6}-\tan ^{-1}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 46, normalized size = 1.10 \begin {gather*} \frac {6 \, x^{6} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 4 \, x^{6} - \sqrt {x^{6} - 1} {\left (4 \, x^{3} - 1\right )}}{6 \, x^{6}} \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 \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 37, normalized size = 0.88
method | result | size |
risch | \(-\frac {4 x^{9}-x^{6}-4 x^{3}+1}{6 x^{6} \sqrt {x^{6}-1}}-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{2}\) | \(37\) |
trager | \(-\frac {\left (4 x^{3}-1\right ) \sqrt {x^{6}-1}}{6 x^{6}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{2}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{7}}\,{d x} \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 \frac {\left (x^3-1\right )\,\sqrt {x^6-1}}{x^7\,\left (x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{7} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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