Optimal. Leaf size=50 \[ \frac {\sqrt {x^6-1}}{3}+\frac {2}{3} \log \left (\sqrt {x^6-1}+x^3\right )+\frac {2}{3} \tan ^{-1}\left (\sqrt {x^6-1}+x^3\right ) \]
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Rubi [F] time = 0.53, 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 \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 \left (-1+x^3\right )} \, dx &=\int \left (\frac {2 \sqrt {-1+x^6}}{3 (-1+x)}-\frac {\sqrt {-1+x^6}}{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-\int \frac {\sqrt {-1+x^6}}{x} \, dx\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^6\right )\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 {1}{3} \sqrt {-1+x^6}+\frac {1}{6} \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 {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 {1}{3} \sqrt {-1+x^6}+\frac {1}{3} \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 {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 {1}{3} \sqrt {-1+x^6}+\frac {1}{3} \tan ^{-1}\left (\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 = 0.13, size = 55, normalized size = 1.10 \begin {gather*} \frac {1}{3} \left (\tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {x^6+2 \sqrt {x^6-1} \log \left (\sqrt {x^6-1}+x^3\right )-1}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 54, normalized size = 1.08 \begin {gather*} \frac {1}{3} \sqrt {-1+x^6}-\frac {2}{3} \tan ^{-1}\left (x^3-\sqrt {-1+x^6}\right )-\frac {2}{3} \log \left (-x^3+\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 42, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} + \frac {2}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - \frac {2}{3} \, \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 \frac {\sqrt {x^{6} - 1} {\left (x^{3} + 1\right )}}{{\left (x^{3} - 1\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 52, normalized size = 1.04
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}}{3}+\frac {2 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(52\) |
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}\,{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\,\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 \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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