Optimal. Leaf size=96 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+x^2}}{\sqrt {x^6+x^2}-x^2}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^6+x^2}}{\sqrt {2}}}{x \sqrt [4]{x^6+x^2}}\right )}{\sqrt {2}} \]
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Rubi [C] time = 0.88, antiderivative size = 319, normalized size of antiderivative = 3.32, number of steps used = 15, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2056, 6715, 6728, 245, 1438, 429, 510} \begin {gather*} -\frac {2 \sqrt [4]{x^4+1} x F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 x^4}{1-i \sqrt {3}},-x^4\right )}{\sqrt [4]{x^6+x^2}}-\frac {2 \sqrt [4]{x^4+1} x F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 x^4}{1+i \sqrt {3}},-x^4\right )}{\sqrt [4]{x^6+x^2}}-\frac {2 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+1} x^3 F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{5 \left (\sqrt {3}+i\right ) \sqrt [4]{x^6+x^2}}-\frac {2 \left (\sqrt {3}+i\right ) \sqrt [4]{x^4+1} x^3 F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{5 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^6+x^2}}+\frac {2 \sqrt [4]{x^4+1} x \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 245
Rule 429
Rule 510
Rule 1438
Rule 2056
Rule 6715
Rule 6728
Rubi steps
\begin {align*} \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {-1+x^4}{\sqrt {x} \sqrt [4]{1+x^4} \left (1+x^2+x^4\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {-1+x^8}{\sqrt [4]{1+x^8} \left (1+x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{1+x^8}}-\frac {2+x^4}{\sqrt [4]{1+x^8} \left (1+x^4+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {2+x^4}{\sqrt [4]{1+x^8} \left (1+x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1-i \sqrt {3}}{\left (1-i \sqrt {3}+2 x^4\right ) \sqrt [4]{1+x^8}}+\frac {1+i \sqrt {3}}{\left (1+i \sqrt {3}+2 x^4\right ) \sqrt [4]{1+x^8}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x^4\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x^4\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {i+\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^8\right ) \sqrt [4]{1+x^8}}+\frac {x^4}{\sqrt [4]{1+x^8} \left (1+i \sqrt {3}+2 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {-i+\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^8\right ) \sqrt [4]{1+x^8}}+\frac {x^4}{\sqrt [4]{1+x^8} \left (1-i \sqrt {3}+2 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{1+x^8} \left (1+i \sqrt {3}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{1+x^8} \left (1-i \sqrt {3}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {2 x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 x^4}{1-i \sqrt {3}},-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {2 x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};-\frac {2 x^4}{1+i \sqrt {3}},-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {2 \left (i-\sqrt {3}\right ) x^3 \sqrt [4]{1+x^4} F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{5 \left (i+\sqrt {3}\right ) \sqrt [4]{x^2+x^6}}-\frac {2 \left (i+\sqrt {3}\right ) x^3 \sqrt [4]{1+x^4} F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{5 \left (i-\sqrt {3}\right ) \sqrt [4]{x^2+x^6}}+\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}\\ \end {align*}
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Mathematica [F] time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 96, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^2+x^6}}{-x^2+\sqrt {x^2+x^6}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2}}}{x \sqrt [4]{x^2+x^6}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 57.21, size = 678, normalized size = 7.06 \begin {gather*} -\frac {1}{2} \, \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}{2} \, \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}{8} \, \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}{8} \, \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 ) \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 {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.00, size = 238, normalized size = 2.48
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-2 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x +2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right ) x \left (x^{2}-x +1\right )}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-2 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x +\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right ) x \left (x^{2}-x +1\right )}\right )}{2}\) | \(238\) |
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 {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}}\,{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 {x^4-1}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^4+x^2+1\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 - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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