Optimal. Leaf size=136 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^6+x^2}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+x^2}}{\sqrt {x^6+x^2}-x^2}\right )}{2 \sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^6+x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^6+x^2}}{\sqrt {2}}}{x \sqrt [4]{x^6+x^2}}\right )}{2 \sqrt {2}} \]
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Rubi [C] time = 0.43, antiderivative size = 163, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2056, 6728, 466, 510} \begin {gather*} \frac {2 \left (-\sqrt {3}+i\right ) x \sqrt [4]{x^6+x^2} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (\sqrt {3}+i\right ) \sqrt [4]{x^4+1}}+\frac {2 \left (\sqrt {3}+i\right ) x \sqrt [4]{x^6+x^2} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 466
Rule 510
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx &=\frac {\sqrt [4]{x^2+x^6} \int \frac {\sqrt {x} \left (-1+x^4\right ) \sqrt [4]{1+x^4}}{1+x^4+x^8} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\sqrt [4]{x^2+x^6} \int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{1-i \sqrt {3}+2 x^4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{1+i \sqrt {3}+2 x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{1+i \sqrt {3}+2 x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{1-i \sqrt {3}+2 x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {2 \left (i-\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (i+\sqrt {3}\right ) \sqrt [4]{1+x^4}}+\frac {2 \left (i+\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (i-\sqrt {3}\right ) \sqrt [4]{1+x^4}}\\ \end {align*}
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Mathematica [F] time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 136, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^6+x^2}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+x^2}}{\sqrt {x^6+x^2}-x^2}\right )}{2 \sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^6+x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^6+x^2}}{\sqrt {2}}}{x \sqrt [4]{x^6+x^2}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 12.43, size = 780, normalized size = 5.74 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {x^{9} + 2 \, x^{7} + 3 \, x^{5} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 3 \, x^{2} + 1\right )} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} + x^{2}} {\left (x^{5} + x^{3} + x\right )} + {\left (16 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} + 2 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 3 \, x^{3} + x\right )} + \sqrt {2} {\left (x^{9} - 8 \, x^{7} + x^{5} - 8 \, x^{3} + x\right )} + 4 \, {\left (x^{6} + x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + x^{3} + x}} + x}{x^{9} - 14 \, x^{7} + 3 \, x^{5} - 14 \, x^{3} + x}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {x^{9} + 2 \, x^{7} + 3 \, x^{5} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} + x^{2}} {\left (x^{5} + x^{3} + x\right )} + {\left (16 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} - 2 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 3 \, x^{3} + x\right )} - \sqrt {2} {\left (x^{9} - 8 \, x^{7} + x^{5} - 8 \, x^{3} + x\right )} + 4 \, {\left (x^{6} + x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + x^{3} + x}} + x}{x^{9} - 14 \, x^{7} + 3 \, x^{5} - 14 \, x^{3} + x}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + x^{3} + x}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x^{2}} x - 2 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + x^{3} + x}\right ) + \frac {1}{4} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x^{5} - x^{3} + x}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} + x^{2}} x + x - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - x^{3} + x}\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^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 10.49, size = 488, normalized size = 3.59 \begin {gather*} -\frac {\ln \left (-\frac {x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+2 \sqrt {x^{6}+x^{2}}\, x +2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+x^{3}+x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{6}+x^{2}}\, x +2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x}{x \left (x^{2}-x +1\right ) \left (x^{2}+x +1\right )}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{2}}\, x -2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x}{x \left (x^{2}-x +1\right ) \left (x^{2}+x +1\right )}\right )}{4} \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^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1}\,{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+x^2\right )}^{1/4}\,\left (x^4-1\right )}{x^8+x^4+1} \,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 {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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