Optimal. Leaf size=67 \[ -\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, antiderivative size = 58, normalized size of antiderivative = 0.87, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1493, 525, 524}
\begin {gather*} -\frac {x^5 \sqrt [4]{1-x^4} \, _2F_1\left (\frac {5}{4},\frac {5}{4};\frac {9}{4};\frac {2 x^4}{x^4+1}\right )}{5 \sqrt [4]{x^4-1} \left (x^4+1\right )^{5/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 524
Rule 525
Rule 1493
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx &=\int \frac {x^4}{\left (-1+x^4\right )^{5/4} \left (1+x^4\right )} \, dx\\ &=-\frac {\sqrt [4]{1-x^4} \int \frac {x^4}{\left (1-x^4\right )^{5/4} \left (1+x^4\right )} \, dx}{\sqrt [4]{-1+x^4}}\\ &=-\frac {x^5 \sqrt [4]{1-x^4} \, _2F_1\left (\frac {5}{4},\frac {5}{4};\frac {9}{4};\frac {2 x^4}{1+x^4}\right )}{5 \sqrt [4]{-1+x^4} \left (1+x^4\right )^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 63, normalized size = 0.94 \begin {gather*} \frac {1}{8} \left (-\frac {4 x}{\sqrt [4]{-1+x^4}}+2^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )+2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.83, size = 216, normalized size = 3.22
method | result | size |
trager | \(-\frac {x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-\sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 x^{4} \RootOf \left (\textit {\_Z}^{4}-8\right )+4 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}-8\right )}{x^{4}+1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}+2 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-4 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}+1}\right )}{16}\) | \(216\) |
risch | \(-\frac {x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}+2 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-4 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}+1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+3 x^{4} \RootOf \left (\textit {\_Z}^{4}-8\right )+4 \left (x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-8\right )}{x^{4}+1}\right )}{16}\) | \(217\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 242 vs.
\(2 (47) = 94\).
time = 3.04, size = 242, normalized size = 3.61 \begin {gather*} -\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{4} - 1\right )} \arctan \left (\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {x^{4} - 1} x^{2} + 2^{\frac {1}{4}} {\left (3 \, x^{4} - 1\right )}\right )}}{2 \, {\left (x^{4} + 1\right )}}\right ) - 2^{\frac {3}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{4} - 1\right )} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) + 2^{\frac {3}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{4} - 1\right )} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) + 16 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{4} - 1\right )}} \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]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \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*} \text {could not integrate} \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^4-1\right )}^{1/4}\,\left (x^8-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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