Optimal. Leaf size=29 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^6-2 x^2+1}}\right )}{\sqrt {2}} \]
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Rubi [F] time = 1.55, 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 {-1+2 x^6}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+2 x^6}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx &=\int \left (\frac {2}{\sqrt {1-2 x^2+x^6}}-\frac {3}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-3 \int \frac {1}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx\\ &=2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-3 \int \left (\frac {1}{3 \left (1+x^2\right ) \sqrt {1-2 x^2+x^6}}+\frac {2-x^2}{3 \left (1-x^2+x^4\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^2+x^6}} \, dx-\int \frac {2-x^2}{\left (1-x^2+x^4\right ) \sqrt {1-2 x^2+x^6}} \, dx\\ &=2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\int \left (\frac {i}{2 (i-x) \sqrt {1-2 x^2+x^6}}+\frac {i}{2 (i+x) \sqrt {1-2 x^2+x^6}}\right ) \, dx-\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx\\ &=-\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}} \, dx-\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}} \, dx\\ &=-\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\left (-1-i \sqrt {3}\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}+\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx-\left (-1+i \sqrt {3}\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}+\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx\\ &=-\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+2 x^6}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.07, size = 29, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-2 x^2+x^6}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 36, normalized size = 1.24 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{6} - 2 \, x^{2} + 1} x}{x^{6} - 4 \, x^{2} + 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 {2 \, x^{6} - 1}{\sqrt {x^{6} - 2 \, x^{2} + 1} {\left (x^{6} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 74, normalized size = 2.55
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{6}-2 x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{4}\) | \(74\) |
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 {2 \, x^{6} - 1}{\sqrt {x^{6} - 2 \, x^{2} + 1} {\left (x^{6} + 1\right )}}\,{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 {2\,x^6-1}{\left (x^6+1\right )\,\sqrt {x^6-2\,x^2+1}} \,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 {2 x^{6} - 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{4} + x^{2} - 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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