Optimal. Leaf size=30 \[ \frac {\sqrt {1-x^6}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {1-x^6}}\right ) \]
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Rubi [F] time = 0.84, 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 {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx &=\int \left (-\frac {\sqrt {1-x^6}}{x^2}+\frac {\left (-1+3 x^4\right ) \sqrt {1-x^6}}{-1-x^2+x^6}\right ) \, dx\\ &=-\int \frac {\sqrt {1-x^6}}{x^2} \, dx+\int \frac {\left (-1+3 x^4\right ) \sqrt {1-x^6}}{-1-x^2+x^6} \, dx\\ &=\frac {\sqrt {1-x^6}}{x}+3 \int \frac {x^4}{\sqrt {1-x^6}} \, dx+\int \left (\frac {\sqrt {1-x^6}}{1+x^2-x^6}+\frac {3 x^4 \sqrt {1-x^6}}{-1-x^2+x^6}\right ) \, dx\\ &=\frac {\sqrt {1-x^6}}{x}-\frac {3}{2} \int \frac {-1+\sqrt {3}-2 x^4}{\sqrt {1-x^6}} \, dx+3 \int \frac {x^4 \sqrt {1-x^6}}{-1-x^2+x^6} \, dx-\frac {1}{2} \left (3 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1-x^6}} \, dx+\int \frac {\sqrt {1-x^6}}{1+x^2-x^6} \, dx\\ &=\frac {\sqrt {1-x^6}}{x}+\frac {3 \left (1+\sqrt {3}\right ) x \sqrt {1-x^6}}{2 \left (1-\left (1+\sqrt {3}\right ) x^2\right )}-\frac {3 \sqrt [4]{3} x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}-\frac {3^{3/4} \left (1-\sqrt {3}\right ) x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}+3 \int \frac {x^4 \sqrt {1-x^6}}{-1-x^2+x^6} \, dx+\int \frac {\sqrt {1-x^6}}{1+x^2-x^6} \, dx\\ \end {align*}
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Mathematica [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.18, size = 30, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-x^6}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {1-x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 42, normalized size = 1.40 \begin {gather*} -\frac {x \arctan \left (\frac {2 \, \sqrt {-x^{6} + 1} x}{x^{6} + x^{2} - 1}\right ) - 2 \, \sqrt {-x^{6} + 1}}{2 \, x} \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 {{\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{{\left (x^{6} - x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.90, size = 78, normalized size = 2.60
method | result | size |
trager | \(\frac {\sqrt {-x^{6}+1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{6}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{2}-1}\right )}{2}\) | \(78\) |
risch | \(-\frac {x^{6}-1}{x \sqrt {-x^{6}+1}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \sqrt {-x^{6}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{2}-1}\right )}{2}\) | \(84\) |
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 {{\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{{\left (x^{6} - x^{2} - 1\right )} x^{2}}\,{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 = -0.03 \begin {gather*} \int -\frac {\sqrt {1-x^6}\,\left (2\,x^6+1\right )}{x^2\,\left (-x^6+x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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