Optimal. Leaf size=26 \[ \frac {\sqrt {x^6-1}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {x^6-1}}\right ) \]
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Rubi [F] time = 0.77, 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 = 0.24, 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 = 3.63, size = 26, 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 = 0.84, size = 40, normalized size = 1.54 \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 = 0.88, size = 71, normalized size = 2.73
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{6}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\) | \(71\) |
risch | \(\frac {\sqrt {x^{6}-1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{6}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\) | \(72\) |
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 = -0.04 \begin {gather*} \int \frac {\sqrt {x^6-1}\,\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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