Optimal. Leaf size=95 \[ \frac {2 \left (x^6+1\right )^{3/4}}{3 x^3}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}-x^2}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+1}}{\sqrt {x^6+1}+x^2}\right ) \]
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Rubi [F] time = 1.03, 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 )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, 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 )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx &=\int \left (-\frac {2}{\sqrt [4]{1+x^6}}-\frac {2}{x^4 \sqrt [4]{1+x^6}}+\frac {x^2}{\sqrt [4]{1+x^6}}+\frac {2 \left (3+x^4\right )}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt [4]{1+x^6}} \, dx\right )-2 \int \frac {1}{x^4 \sqrt [4]{1+x^6}} \, dx+2 \int \frac {3+x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx+\int \frac {x^2}{\sqrt [4]{1+x^6}} \, dx\\ &=-2 x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2}} \, dx,x,x^3\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1+x^2}} \, dx,x,x^3\right )+2 \int \left (\frac {3}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )}+\frac {x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )}\right ) \, dx\\ &=\frac {2 x^3}{3 \sqrt [4]{1+x^6}}+\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}-2 x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx,x,x^3\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^2}} \, dx,x,x^3\right )+2 \int \frac {x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx\\ &=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}-\frac {2}{3} E\left (\left .\frac {1}{2} \tan ^{-1}\left (x^3\right )\right |2\right )-2 x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx,x,x^3\right )+2 \int \frac {x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx\\ &=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}-2 x \, _2F_1\left (\frac {1}{6},\frac {1}{4};\frac {7}{6};-x^6\right )+2 \int \frac {x^4}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx+6 \int \frac {1}{\sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^4 \sqrt [4]{1+x^6} \left (1+x^4+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 7.03, size = 95, normalized size = 1.00 \begin {gather*} \frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 173.50, size = 702, normalized size = 7.39 \begin {gather*} \frac {12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{12} + 2 \, x^{10} + x^{8} + 2 \, x^{6} + 2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{5} + x\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} - x^{7} + 3 \, x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} + x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} + {\left (16 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (x^{8} - 3 \, x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} + \sqrt {2} {\left (x^{12} - 8 \, x^{10} - x^{8} + 2 \, x^{6} - 8 \, x^{4} + 1\right )} + 4 \, {\left (x^{9} + x^{7} + x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} + x^{4} + 1}} + 1}{x^{12} - 14 \, x^{10} + x^{8} + 2 \, x^{6} - 14 \, x^{4} + 1}\right ) - 12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{12} + 2 \, x^{10} + x^{8} + 2 \, x^{6} + 2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{5} + x\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} - x^{7} + 3 \, x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} + x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} + {\left (16 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (x^{8} - 3 \, x^{6} + x^{2}\right )} \sqrt {x^{6} + 1} - \sqrt {2} {\left (x^{12} - 8 \, x^{10} - x^{8} + 2 \, x^{6} - 8 \, x^{4} + 1\right )} + 4 \, {\left (x^{9} + x^{7} + x^{3}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} + x^{4} + 1}} + 1}{x^{12} - 14 \, x^{10} + x^{8} + 2 \, x^{6} - 14 \, x^{4} + 1}\right ) + 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{6} + x^{4} + 1}\right ) - 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{6} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{6} + x^{4} + 1}\right ) + 8 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{12 \, x^{3}} \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} + x^{4} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 31.98, size = 216, normalized size = 2.27
method | result | size |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (x^{6}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{6}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )\) | \(216\) |
risch | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{6}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}+x^{4}+1}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \left (x^{6}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{6}+x^{4}+1}\right )\) | \(217\) |
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} + x^{4} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{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 )}{x^4\,{\left (x^6+1\right )}^{1/4}\,\left (x^6+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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