Optimal. Leaf size=162 \[ -\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{4\ 2^{3/4}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{4\ 2^{3/4}} \]
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Rubi [C] time = 0.64, antiderivative size = 127, normalized size of antiderivative = 0.78, 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, 6725, 245, 1438, 429, 510} \begin {gather*} -\frac {4 \sqrt [4]{x^4+1} x F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^6+x^2}}-\frac {2 \sqrt [4]{x^4+1} x^3 F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};x^4,-x^4\right )}{5 \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 6725
Rubi steps
\begin {align*} \int \frac {1+x^2+x^4}{\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {1+x^2+x^4}{\sqrt {x} \left (-1+x^4\right ) \sqrt [4]{1+x^4}} \, 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^4+x^8}{\left (-1+x^8\right ) \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 \left (\frac {1}{\sqrt [4]{1+x^8}}+\frac {2+x^4}{\left (-1+x^8\right ) \sqrt [4]{1+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 \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}{\left (-1+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} \, _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 {3}{2 \left (1-x^4\right ) \sqrt [4]{1+x^8}}-\frac {1}{2 \left (1+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 (\sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^4\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (3 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-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 (\sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (1-x^8\right ) \sqrt [4]{1+x^8}}+\frac {x^4}{\left (-1+x^8\right ) \sqrt [4]{1+x^8}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (3 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (1-x^8\right ) \sqrt [4]{1+x^8}}-\frac {x^4}{\left (-1+x^8\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 (\sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (3 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (3 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {4 x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {2 x^3 \sqrt [4]{1+x^4} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};x^4,-x^4\right )}{5 \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 [C] time = 0.20, size = 95, normalized size = 0.59 \begin {gather*} -\frac {2 \sqrt [4]{x^4+1} \left (45 x F_1\left (\frac {1}{8};\frac {1}{4},1;\frac {9}{8};-x^4,x^4\right )+5 x^5 F_1\left (\frac {9}{8};\frac {1}{4},1;\frac {17}{8};-x^4,x^4\right )+9 x^3 F_1\left (\frac {5}{8};\frac {1}{4},1;\frac {13}{8};-x^4,x^4\right )\right )}{45 \sqrt [4]{x^6+x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.80, size = 162, normalized size = 1.00 \begin {gather*} -\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{4\ 2^{3/4}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{4\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 101.19, size = 1002, normalized size = 6.19
result too large to display
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} + x^{2} + 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 22.89, size = 633, normalized size = 3.91
method | result | size |
trager | \(-\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{16}-\frac {3 \RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} \sqrt {x^{6}+x^{2}}\, x +\RootOf \left (\textit {\_Z}^{4}-8\right ) x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-8\right ) x}{\left (-1+x \right )^{2} \left (1+x \right )^{2} x}\right )}{16}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (x^{2}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{3}}{32}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x +2 \sqrt {x^{6}+x^{2}}}{x \left (x^{2}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2}}{32}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{5}-\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{5}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{3}+8 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{2}-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \sqrt {x^{6}+x^{2}}\, x -8 \sqrt {x^{6}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}-8\right ) x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x -\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x +16 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )^{2}}\right )}{32}\) | \(633\) |
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} + x^{2} + 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 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+x^2+1}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\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^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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