Optimal. Leaf size=119 \[ \frac {2 \sqrt [4]{x^6-x^4-2}}{x}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-x^4-2}}{\sqrt {x^6-x^4-2}-x^2}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-x^4-2}}{x^2+\sqrt {x^6-x^4-2}}\right )}{\sqrt {2}} \]
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Rubi [F] time = 1.75, 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 (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-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 (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx &=\int \left (-\frac {2 \sqrt [4]{-2-x^4+x^6}}{x^2}+\frac {3 x^4 \sqrt [4]{-2-x^4+x^6}}{-2+x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-2-x^4+x^6}}{x^2} \, dx\right )+3 \int \frac {x^4 \sqrt [4]{-2-x^4+x^6}}{-2+x^6} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-2-x^4+x^6}}{x^2} \, dx\right )+3 \int \left (\frac {x \sqrt [4]{-2-x^4+x^6}}{2 \left (-\sqrt {2}+x^3\right )}+\frac {x \sqrt [4]{-2-x^4+x^6}}{2 \left (\sqrt {2}+x^3\right )}\right ) \, dx\\ &=\frac {3}{2} \int \frac {x \sqrt [4]{-2-x^4+x^6}}{-\sqrt {2}+x^3} \, dx+\frac {3}{2} \int \frac {x \sqrt [4]{-2-x^4+x^6}}{\sqrt {2}+x^3} \, dx-2 \int \frac {\sqrt [4]{-2-x^4+x^6}}{x^2} \, dx\\ &=\frac {3}{2} \int \left (-\frac {\sqrt [4]{-2-x^4+x^6}}{3 \sqrt [6]{2} \left (\sqrt [6]{2}-x\right )}-\frac {(-1)^{2/3} \sqrt [4]{-2-x^4+x^6}}{3 \sqrt [6]{2} \left (\sqrt [6]{2}+\sqrt [3]{-1} x\right )}+\frac {\sqrt [3]{-1} \sqrt [4]{-2-x^4+x^6}}{3 \sqrt [6]{2} \left (\sqrt [6]{2}-(-1)^{2/3} x\right )}\right ) \, dx+\frac {3}{2} \int \left (-\frac {\sqrt [4]{-2-x^4+x^6}}{3 \sqrt [6]{2} \left (\sqrt [6]{2}+x\right )}-\frac {(-1)^{2/3} \sqrt [4]{-2-x^4+x^6}}{3 \sqrt [6]{2} \left (\sqrt [6]{2}-\sqrt [3]{-1} x\right )}+\frac {\sqrt [3]{-1} \sqrt [4]{-2-x^4+x^6}}{3 \sqrt [6]{2} \left (\sqrt [6]{2}+(-1)^{2/3} x\right )}\right ) \, dx-2 \int \frac {\sqrt [4]{-2-x^4+x^6}}{x^2} \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{-2-x^4+x^6}}{x^2} \, dx\right )-\frac {\int \frac {\sqrt [4]{-2-x^4+x^6}}{\sqrt [6]{2}-x} \, dx}{2 \sqrt [6]{2}}-\frac {\int \frac {\sqrt [4]{-2-x^4+x^6}}{\sqrt [6]{2}+x} \, dx}{2 \sqrt [6]{2}}+\frac {\sqrt [3]{-1} \int \frac {\sqrt [4]{-2-x^4+x^6}}{\sqrt [6]{2}-(-1)^{2/3} x} \, dx}{2 \sqrt [6]{2}}+\frac {\sqrt [3]{-1} \int \frac {\sqrt [4]{-2-x^4+x^6}}{\sqrt [6]{2}+(-1)^{2/3} x} \, dx}{2 \sqrt [6]{2}}-\frac {(-1)^{2/3} \int \frac {\sqrt [4]{-2-x^4+x^6}}{\sqrt [6]{2}-\sqrt [3]{-1} x} \, dx}{2 \sqrt [6]{2}}-\frac {(-1)^{2/3} \int \frac {\sqrt [4]{-2-x^4+x^6}}{\sqrt [6]{2}+\sqrt [3]{-1} x} \, dx}{2 \sqrt [6]{2}}\\ \end {align*}
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Mathematica [F] time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.03, size = 119, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{x^6-x^4-2}}{x}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-x^4-2}}{\sqrt {x^6-x^4-2}-x^2}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-x^4-2}}{x^2+\sqrt {x^6-x^4-2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 101.29, size = 780, normalized size = 6.55 \begin {gather*} -\frac {4 \, \sqrt {2} x \arctan \left (-\frac {x^{12} - 4 \, x^{6} + 2 \, \sqrt {2} {\left (x^{7} - 4 \, x^{5} - 2 \, x\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} - 4 \, x^{7} - 6 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} - {\left (16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (x^{8} - 4 \, x^{6} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} + \sqrt {2} {\left (x^{12} - 10 \, x^{10} + 8 \, x^{8} - 4 \, x^{6} + 20 \, x^{4} + 4\right )} + 4 \, {\left (x^{9} - 2 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2}{x^{6} - 2}} + 4}{x^{12} - 16 \, x^{10} + 16 \, x^{8} - 4 \, x^{6} + 32 \, x^{4} + 4}\right ) - 4 \, \sqrt {2} x \arctan \left (-\frac {x^{12} - 4 \, x^{6} - 2 \, \sqrt {2} {\left (x^{7} - 4 \, x^{5} - 2 \, x\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} - 4 \, x^{7} - 6 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} - {\left (16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (x^{8} - 4 \, x^{6} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} - \sqrt {2} {\left (x^{12} - 10 \, x^{10} + 8 \, x^{8} - 4 \, x^{6} + 20 \, x^{4} + 4\right )} + 4 \, {\left (x^{9} - 2 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2}{x^{6} - 2}} + 4}{x^{12} - 16 \, x^{10} + 16 \, x^{8} - 4 \, x^{6} + 32 \, x^{4} + 4}\right ) + \sqrt {2} x \log \left (\frac {4 \, {\left (x^{6} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2\right )}}{x^{6} - 2}\right ) - \sqrt {2} x \log \left (\frac {4 \, {\left (x^{6} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2\right )}}{x^{6} - 2}\right ) - 16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}}{8 \, 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 (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )}}{{\left (x^{6} - 2\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.81, size = 1372, normalized size = 11.53
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} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )}}{{\left (x^{6} - 2\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 = -0.01 \begin {gather*} \int \frac {\left (x^6+4\right )\,{\left (x^6-x^4-2\right )}^{1/4}}{x^2\,\left (x^6-2\right )} \,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 {\left (x^{6} + 4\right ) \sqrt [4]{x^{6} - x^{4} - 2}}{x^{2} \left (x^{6} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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