Optimal. Leaf size=141 \[ -\frac {\left (-x^6-x^4+1\right )^{3/4} x}{2 \left (x^6-1\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x^6-x^4+1}}{\sqrt {-x^6-x^4+1}-x^2}\right )}{4 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x^6-x^4+1}}{x^2+\sqrt {-x^6-x^4+1}}\right )}{4 \sqrt {2}} \]
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Rubi [F] time = 2.74, 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^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx &=\int \left (\frac {1}{\sqrt [4]{1-x^4-x^6}}-\frac {1}{12 (-1+x)^2 \sqrt [4]{1-x^4-x^6}}-\frac {1}{12 (1+x)^2 \sqrt [4]{1-x^4-x^6}}+\frac {5}{6 \left (-1+x^2\right ) \sqrt [4]{1-x^4-x^6}}+\frac {1}{4 \left (1-x+x^2\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {5 (-3+x)}{12 \left (1-x+x^2\right ) \sqrt [4]{1-x^4-x^6}}+\frac {1}{4 \left (1+x+x^2\right )^2 \sqrt [4]{1-x^4-x^6}}-\frac {5 (3+x)}{12 \left (1+x+x^2\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {1}{(-1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx\right )-\frac {1}{12} \int \frac {1}{(1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{4} \int \frac {1}{\left (1-x+x^2\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{4} \int \frac {1}{\left (1+x+x^2\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {5}{12} \int \frac {-3+x}{\left (1-x+x^2\right ) \sqrt [4]{1-x^4-x^6}} \, dx-\frac {5}{12} \int \frac {3+x}{\left (1+x+x^2\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\frac {5}{6} \int \frac {1}{\left (-1+x^2\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\int \frac {1}{\sqrt [4]{1-x^4-x^6}} \, dx\\ &=-\left (\frac {1}{12} \int \frac {1}{(-1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx\right )-\frac {1}{12} \int \frac {1}{(1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{4} \int \left (-\frac {4}{3 \left (1+i \sqrt {3}-2 x\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {4 i}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right ) \sqrt [4]{1-x^4-x^6}}-\frac {4}{3 \left (-1+i \sqrt {3}+2 x\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {4 i}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx+\frac {1}{4} \int \left (-\frac {4}{3 \left (-1+i \sqrt {3}-2 x\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {4 i}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right ) \sqrt [4]{1-x^4-x^6}}-\frac {4}{3 \left (1+i \sqrt {3}+2 x\right )^2 \sqrt [4]{1-x^4-x^6}}+\frac {4 i}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx+\frac {5}{12} \int \left (\frac {1+\frac {5 i}{\sqrt {3}}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}+\frac {1-\frac {5 i}{\sqrt {3}}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx-\frac {5}{12} \int \left (\frac {1-\frac {5 i}{\sqrt {3}}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}+\frac {1+\frac {5 i}{\sqrt {3}}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}}\right ) \, dx+\frac {5}{6} \int \left (\frac {1}{2 (-1+x) \sqrt [4]{1-x^4-x^6}}-\frac {1}{2 (1+x) \sqrt [4]{1-x^4-x^6}}\right ) \, dx+\int \frac {1}{\sqrt [4]{1-x^4-x^6}} \, dx\\ &=-\left (\frac {1}{12} \int \frac {1}{(-1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx\right )-\frac {1}{12} \int \frac {1}{(1+x)^2 \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{3} \int \frac {1}{\left (-1+i \sqrt {3}-2 x\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+i \sqrt {3}-2 x\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{3} \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{3} \int \frac {1}{\left (1+i \sqrt {3}+2 x\right )^2 \sqrt [4]{1-x^4-x^6}} \, dx+\frac {5}{12} \int \frac {1}{(-1+x) \sqrt [4]{1-x^4-x^6}} \, dx-\frac {5}{12} \int \frac {1}{(1+x) \sqrt [4]{1-x^4-x^6}} \, dx+\frac {i \int \frac {1}{\left (-1+i \sqrt {3}-2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx}{3 \sqrt {3}}+\frac {i \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx}{3 \sqrt {3}}+\frac {i \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx}{3 \sqrt {3}}+\frac {i \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx}{3 \sqrt {3}}-\frac {1}{36} \left (5 \left (3-5 i \sqrt {3}\right )\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{36} \left (5 \left (3-5 i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\frac {1}{36} \left (5 \left (3+5 i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx-\frac {1}{36} \left (5 \left (3+5 i \sqrt {3}\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1-x^4-x^6}} \, dx+\int \frac {1}{\sqrt [4]{1-x^4-x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 1.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+x^6\right ) \left (-1-x^4+x^6\right )}{\sqrt [4]{1-x^4-x^6} \left (-1+x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.88, size = 141, normalized size = 1.00 \begin {gather*} -\frac {\left (-x^6-x^4+1\right )^{3/4} x}{2 \left (x^6-1\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x^6-x^4+1}}{\sqrt {-x^6-x^4+1}-x^2}\right )}{4 \sqrt {2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-x^6-x^4+1}}{x^2+\sqrt {-x^6-x^4+1}}\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 10.15, size = 852, normalized size = 6.04 \begin {gather*} -\frac {20 \, \sqrt {2} {\left (x^{6} - 1\right )} \arctan \left (-\frac {x^{12} - 2 \, x^{6} + 2 \, \sqrt {2} {\left (x^{7} + 4 \, x^{5} - x\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} + 4 \, x^{7} - 3 \, x^{3}\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} - 4 \, {\left (x^{8} - x^{2}\right )} \sqrt {-x^{6} - x^{4} + 1} - {\left (16 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} - x^{2}\right )} \sqrt {-x^{6} - x^{4} + 1} - \sqrt {2} {\left (x^{12} + 10 \, x^{10} + 8 \, x^{8} - 2 \, x^{6} - 10 \, x^{4} + 1\right )} - 4 \, {\left (x^{9} - x^{3}\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {-x^{6} - x^{4} + 1} x^{2} + 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x - 1}{x^{6} - 1}} + 1}{x^{12} + 16 \, x^{10} + 16 \, x^{8} - 2 \, x^{6} - 16 \, x^{4} + 1}\right ) - 20 \, \sqrt {2} {\left (x^{6} - 1\right )} \arctan \left (-\frac {x^{12} - 2 \, x^{6} - 2 \, \sqrt {2} {\left (x^{7} + 4 \, x^{5} - x\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} + 4 \, x^{7} - 3 \, x^{3}\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} - 4 \, {\left (x^{8} - x^{2}\right )} \sqrt {-x^{6} - x^{4} + 1} - {\left (16 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} - x^{2}\right )} \sqrt {-x^{6} - x^{4} + 1} + \sqrt {2} {\left (x^{12} + 10 \, x^{10} + 8 \, x^{8} - 2 \, x^{6} - 10 \, x^{4} + 1\right )} - 4 \, {\left (x^{9} - x^{3}\right )} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} - 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {-x^{6} - x^{4} + 1} x^{2} - 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x - 1}{x^{6} - 1}} + 1}{x^{12} + 16 \, x^{10} + 16 \, x^{8} - 2 \, x^{6} - 16 \, x^{4} + 1}\right ) - 5 \, \sqrt {2} {\left (x^{6} - 1\right )} \log \left (\frac {4 \, {\left (x^{6} + 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {-x^{6} - x^{4} + 1} x^{2} + 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{6} - 1}\right ) + 5 \, \sqrt {2} {\left (x^{6} - 1\right )} \log \left (\frac {4 \, {\left (x^{6} - 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {-x^{6} - x^{4} + 1} x^{2} - 2 \, \sqrt {2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{6} - 1}\right ) + 16 \, {\left (-x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{6} - 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 {{\left (x^{6} - x^{4} - 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{6} - 1\right )}^{2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.94, size = 321, normalized size = 2.28 \begin {gather*} \frac {x \left (x^{6}+x^{4}-1\right )}{2 \left (x^{6}-1\right ) \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}}}-\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {-x^{6}-x^{4}+1}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{8}-\frac {5 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {-x^{6}-x^{4}+1}\, x^{2}+2 \left (-x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{8} \end {gather*}
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 )} {\left (x^{6} + 2\right )}}{{\left (x^{6} - 1\right )}^{2} {\left (-x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}}\,{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+x^4+1\right )}{{\left (x^6-1\right )}^2\,{\left (-x^6-x^4+1\right )}^{1/4}} \,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} + 2\right ) \left (x^{6} - x^{4} - 1\right )}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2} \left (x^{2} + x + 1\right )^{2} \sqrt [4]{- x^{6} - x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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