Optimal. Leaf size=101 \[ \frac {\left (x^4+x^2\right )^{3/4} \left (-13 x^4-2 x^2-5\right )}{80 x \left (x^2-1\right ) \left (x^2+1\right )^2}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^2}}\right )}{32 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^2}}\right )}{32 \sqrt [4]{2}} \]
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Rubi [C] time = 9.82, antiderivative size = 109, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2056, 1254, 466, 510} \begin {gather*} \frac {128 x^5 \Gamma \left (\frac {13}{4}\right ) \left (17 \left (-4 x^4-9 x^2+13\right ) \, _2F_1\left (1,2;\frac {17}{4};-\frac {2 x^2}{1-x^2}\right )-64 x^2 \left (x^2+1\right ) \, _2F_1\left (2,3;\frac {21}{4};-\frac {2 x^2}{1-x^2}\right )\right )}{89505 \left (1-x^2\right )^3 \left (x^2+1\right ) \sqrt [4]{x^4+x^2} \Gamma \left (\frac {1}{4}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 466
Rule 510
Rule 1254
Rule 2056
Rubi steps
\begin {align*} \int \frac {x^4}{\left (-1+x^4\right )^2 \sqrt [4]{x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{1+x^2} \left (-1+x^4\right )^2} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {x^{7/2}}{\left (-1+x^2\right )^2 \left (1+x^2\right )^{9/4}} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-1+x^4\right )^2 \left (1+x^4\right )^{9/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}\\ &=\frac {128 x^5 \Gamma \left (\frac {13}{4}\right ) \left (17 \left (13-9 x^2-4 x^4\right ) \, _2F_1\left (1,2;\frac {17}{4};-\frac {2 x^2}{1-x^2}\right )-64 x^2 \left (1+x^2\right ) \, _2F_1\left (2,3;\frac {21}{4};-\frac {2 x^2}{1-x^2}\right )\right )}{89505 \left (1-x^2\right )^3 \left (1+x^2\right ) \sqrt [4]{x^2+x^4} \Gamma \left (\frac {1}{4}\right )}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 112, normalized size = 1.11 \begin {gather*} \frac {x \left (-4 \left (13 x^4+2 x^2+5\right )+5\ 2^{3/4} \sqrt [4]{\frac {1}{x^2}+1} \left (x^4-1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{2}}\right )-5\ 2^{3/4} \sqrt [4]{\frac {1}{x^2}+1} \left (x^4-1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{x^2}+1}}{\sqrt [4]{2}}\right )\right )}{320 \left (x^4-1\right ) \sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 101, normalized size = 1.00 \begin {gather*} \frac {\left (-5-2 x^2-13 x^4\right ) \left (x^2+x^4\right )^{3/4}}{80 x \left (-1+x^2\right ) \left (1+x^2\right )^2}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{32 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{32 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.81, size = 311, normalized size = 3.08 \begin {gather*} \frac {20 \cdot 2^{\frac {3}{4}} {\left (x^{7} + x^{5} - x^{3} - x\right )} \arctan \left (\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {x^{4} + x^{2}} x + 2^{\frac {1}{4}} {\left (3 \, x^{3} + x\right )}\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{3} - x\right )}}\right ) - 5 \cdot 2^{\frac {3}{4}} {\left (x^{7} + x^{5} - x^{3} - x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{3} + x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) + 5 \cdot 2^{\frac {3}{4}} {\left (x^{7} + x^{5} - x^{3} - x\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{3} + x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} + x^{2}} x + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} - x}\right ) - 16 \, {\left (13 \, x^{4} + 2 \, x^{2} + 5\right )} {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{1280 \, {\left (x^{7} + x^{5} - x^{3} - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 81, normalized size = 0.80 \begin {gather*} \frac {1}{64} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{128} \cdot 2^{\frac {3}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {{\left (\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}}}{16 \, {\left (\frac {1}{x^{2}} - 1\right )}} - \frac {1}{10 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.70, size = 267, normalized size = 2.64
method | result | size |
risch | \(-\frac {x \left (13 x^{4}+2 x^{2}+5\right )}{80 \left (x^{2}-1\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{2}+1\right )}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}+x^{2}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (1+x \right ) x \left (-1+x \right )}\right )}{128}-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\sqrt {x^{4}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x +2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-8\right ) x}{x \left (-1+x \right ) \left (1+x \right )}\right )}{128}\) | \(267\) |
trager | \(-\frac {\left (13 x^{4}+2 x^{2}+5\right ) \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{80 \left (x^{2}-1\right ) \left (x^{2}+1\right )^{2} x}+\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{4}+x^{2}}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x -2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}-4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-8\right ) x}{\left (1+x \right ) x \left (-1+x \right )}\right )}{128}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}+x^{2}}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x}{\left (1+x \right ) x \left (-1+x \right )}\right )}{128}\) | \(268\) |
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*} \frac {2 \, {\left (4 \, x^{5} + x^{3} - 3 \, x\right )} x^{\frac {7}{2}}}{21 \, {\left (x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} + \int \frac {16 \, {\left (4 \, x^{4} + x^{2} - 3\right )} x^{\frac {7}{2}}}{21 \, {\left (x^{12} - 3 \, x^{8} + 3 \, x^{4} - 1\right )} {\left (x^{2} + 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 {x^4}{{\left (x^4+x^2\right )}^{1/4}\,{\left (x^4-1\right )}^2} \,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 {x^{4}}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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