Optimal. Leaf size=104 \[ \frac {1}{4} \sqrt [4]{\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{x^6+2 x^4-2}}\right )-\frac {1}{4} \sqrt [4]{\frac {5}{2}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{x^6+2 x^4-2}}\right )+\frac {\sqrt [4]{x^6+2 x^4-2} \left (2 x^6+9 x^4-4\right )}{10 x^5} \]
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Rubi [F] time = 1.73, 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 (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 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 (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx &=\int \left (\frac {1}{2} \sqrt [4]{-2+2 x^4+x^6}+\frac {2 \sqrt [4]{-2+2 x^4+x^6}}{x^6}-\frac {\sqrt [4]{-2+2 x^4+x^6}}{2 x^2}+\frac {x^2 \left (1-3 x^2\right ) \sqrt [4]{-2+2 x^4+x^6}}{2 \left (4+x^4-2 x^6\right )}\right ) \, dx\\ &=\frac {1}{2} \int \sqrt [4]{-2+2 x^4+x^6} \, dx-\frac {1}{2} \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^2} \, dx+\frac {1}{2} \int \frac {x^2 \left (1-3 x^2\right ) \sqrt [4]{-2+2 x^4+x^6}}{4+x^4-2 x^6} \, dx+2 \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^6} \, dx\\ &=\frac {1}{2} \int \sqrt [4]{-2+2 x^4+x^6} \, dx-\frac {1}{2} \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^2} \, dx+\frac {1}{2} \int \left (-\frac {x^2 \sqrt [4]{-2+2 x^4+x^6}}{-4-x^4+2 x^6}+\frac {3 x^4 \sqrt [4]{-2+2 x^4+x^6}}{-4-x^4+2 x^6}\right ) \, dx+2 \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^6} \, dx\\ &=\frac {1}{2} \int \sqrt [4]{-2+2 x^4+x^6} \, dx-\frac {1}{2} \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^2} \, dx-\frac {1}{2} \int \frac {x^2 \sqrt [4]{-2+2 x^4+x^6}}{-4-x^4+2 x^6} \, dx+\frac {3}{2} \int \frac {x^4 \sqrt [4]{-2+2 x^4+x^6}}{-4-x^4+2 x^6} \, dx+2 \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^6} \, dx\\ \end {align*}
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Mathematica [F] time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.80, size = 104, normalized size = 1.00 \begin {gather*} \frac {1}{4} \sqrt [4]{\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{x^6+2 x^4-2}}\right )-\frac {1}{4} \sqrt [4]{\frac {5}{2}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{x^6+2 x^4-2}}\right )+\frac {\sqrt [4]{x^6+2 x^4-2} \left (2 x^6+9 x^4-4\right )}{10 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 133.64, size = 380, normalized size = 3.65 \begin {gather*} \frac {20 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \arctan \left (\frac {20 \cdot 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 20 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {5} {\left (4 \cdot 5^{\frac {3}{4}} 2^{\frac {1}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} + 5^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )}\right )} \sqrt {\sqrt {5} \sqrt {2}}}{10 \, {\left (2 \, x^{6} - x^{4} - 4\right )}}\right ) - 5 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (-\frac {10 \, \sqrt {5} \sqrt {2} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 10 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} + 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} + 20 \, {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{6} - x^{4} - 4}\right ) + 5 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (-\frac {10 \, \sqrt {5} \sqrt {2} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} - 10 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} - 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} + 20 \, {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{6} - x^{4} - 4}\right ) + 16 \, {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}}}{160 \, 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} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )} {\left (x^{6} - 2\right )}}{{\left (2 \, x^{6} - x^{4} - 4\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.31, size = 1513, normalized size = 14.55
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} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )} {\left (x^{6} - 2\right )}}{{\left (2 \, x^{6} - x^{4} - 4\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^6-2\right )\,\left (x^6+4\right )\,{\left (x^6+2\,x^4-2\right )}^{1/4}}{x^6\,\left (-2\,x^6+x^4+4\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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