Optimal. Leaf size=70 \[ 2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^6-x^4+1}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^6-x^4+1}}\right )+\frac {2 \sqrt [4]{x^6-x^4+1} \left (x^6+9 x^4+1\right )}{5 x^5} \]
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Rubi [F] time = 2.44, 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 (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx &=\int \left (\sqrt [4]{1-x^4+x^6}+\frac {\sqrt [4]{1-x^4+x^6}}{-1-x}+\frac {\sqrt [4]{1-x^4+x^6}}{-1+x}-\frac {2 \sqrt [4]{1-x^4+x^6}}{x^6}-\frac {4 \sqrt [4]{1-x^4+x^6}}{x^2}+\frac {2 \left (-1+2 x^2\right ) \sqrt [4]{1-x^4+x^6}}{-1-x^2+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )+2 \int \frac {\left (-1+2 x^2\right ) \sqrt [4]{1-x^4+x^6}}{-1-x^2+x^4} \, dx-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )+2 \int \left (\frac {2 \sqrt [4]{1-x^4+x^6}}{-1-\sqrt {5}+2 x^2}+\frac {2 \sqrt [4]{1-x^4+x^6}}{-1+\sqrt {5}+2 x^2}\right ) \, dx-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+4 \int \frac {\sqrt [4]{1-x^4+x^6}}{-1-\sqrt {5}+2 x^2} \, dx+4 \int \frac {\sqrt [4]{1-x^4+x^6}}{-1+\sqrt {5}+2 x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+4 \int \left (\frac {i \sqrt [4]{1-x^4+x^6}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {i \sqrt [4]{1-x^4+x^6}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx+4 \int \left (\frac {\sqrt {1+\sqrt {5}} \sqrt [4]{1-x^4+x^6}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {\sqrt {1+\sqrt {5}} \sqrt [4]{1-x^4+x^6}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\frac {(2 i) \int \frac {\sqrt [4]{1-x^4+x^6}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {-1+\sqrt {5}}}+\frac {(2 i) \int \frac {\sqrt [4]{1-x^4+x^6}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {-1+\sqrt {5}}}-\frac {2 \int \frac {\sqrt [4]{1-x^4+x^6}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {1+\sqrt {5}}}-\frac {2 \int \frac {\sqrt [4]{1-x^4+x^6}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {1+\sqrt {5}}}+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx\\ \end {align*}
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Mathematica [F] time = 0.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.60, size = 70, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^6-x^4+1}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^6-x^4+1}}\right )+\frac {2 \sqrt [4]{x^6-x^4+1} \left (x^6+9 x^4+1\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 160.79, size = 154, normalized size = 2.20 \begin {gather*} \frac {5 \, x^{5} \arctan \left (\frac {2 \, {\left ({\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{6} - 2 \, x^{4} + 1}\right ) + 5 \, x^{5} \log \left (\frac {x^{6} - 2 \, {\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{6} - x^{4} + 1} x^{2} - 2 \, {\left (x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} - 2 \, x^{4} + 1}\right ) + 2 \, {\left (x^{6} + 9 \, x^{4} + 1\right )} {\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \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 (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{6} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{6} - 2 \, x^{4} + 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.07, size = 1242, normalized size = 17.74
result too large to display
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 (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{6} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{6} - 2 \, x^{4} + 1\right )} x^{6}}\,{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 = -0.01 \begin {gather*} \int \frac {\left (x^6+1\right )\,\left (x^6-2\right )\,{\left (x^6-x^4+1\right )}^{1/4}}{x^6\,\left (x^6-2\,x^4+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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