Optimal. Leaf size=106 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2 x^4-1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4-1}}\right )}{2^{3/4}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2 x^4-1}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4-1}}\right )}{2^{3/4}}+\frac {\left (2 x^4-1\right )^{5/4}}{5 x^5} \]
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Rubi [C] time = 0.65, antiderivative size = 67, normalized size of antiderivative = 0.63, number of steps used = 23, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6725, 264, 1240, 407, 409, 1213, 537, 511, 510} \begin {gather*} \frac {\left (2 x^4-1\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{2 x^4-1} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{3 \sqrt [4]{1-2 x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 264
Rule 407
Rule 409
Rule 510
Rule 511
Rule 537
Rule 1213
Rule 1240
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-1+2 x^4} \left (-1+x^4+x^8\right )}{x^6 \left (-1+x^4\right )} \, dx &=\int \left (\frac {\sqrt [4]{-1+2 x^4}}{x^6}+\frac {\sqrt [4]{-1+2 x^4}}{2 \left (-1+x^2\right )}+\frac {\sqrt [4]{-1+2 x^4}}{2 \left (1+x^2\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^2} \, dx+\frac {1}{2} \int \frac {\sqrt [4]{-1+2 x^4}}{1+x^2} \, dx+\int \frac {\sqrt [4]{-1+2 x^4}}{x^6} \, dx\\ &=\frac {\left (-1+2 x^4\right )^{5/4}}{5 x^5}+\frac {1}{2} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{1-x^4}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4}\right ) \, dx+\frac {1}{2} \int \left (\frac {\sqrt [4]{-1+2 x^4}}{-1+x^4}+\frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4}\right ) \, dx\\ &=\frac {\left (-1+2 x^4\right )^{5/4}}{5 x^5}+\frac {1}{2} \int \frac {\sqrt [4]{-1+2 x^4}}{1-x^4} \, dx+\frac {1}{2} \int \frac {\sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx+2 \left (\frac {1}{2} \int \frac {x^2 \sqrt [4]{-1+2 x^4}}{-1+x^4} \, dx\right )\\ &=\frac {\left (-1+2 x^4\right )^{5/4}}{5 x^5}+2 \frac {\sqrt [4]{-1+2 x^4} \int \frac {x^2 \sqrt [4]{1-2 x^4}}{-1+x^4} \, dx}{2 \sqrt [4]{1-2 x^4}}+\frac {1}{2} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (1-x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {1}{2} \left (\sqrt {-\frac {1}{-1+2 x^4}} \sqrt {-1+2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x^4} \left (-1+x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-1+2 x^4}}\right )\\ &=\frac {\left (-1+2 x^4\right )^{5/4}}{5 x^5}-\frac {x^3 \sqrt [4]{-1+2 x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )}{3 \sqrt [4]{1-2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 67, normalized size = 0.63 \begin {gather*} \frac {\sqrt [4]{2 x^4-1} \left (-5 x^8 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};2 x^4,x^4\right )-3 \left (1-2 x^4\right )^{5/4}\right )}{15 x^5 \sqrt [4]{1-2 x^4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.32, size = 106, normalized size = 1.00 \begin {gather*} \frac {\left (-1+2 x^4\right )^{5/4}}{5 x^5}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )}{2^{3/4}}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+2 x^4}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+2 x^4}}\right )}{2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 22.76, size = 330, normalized size = 3.11 \begin {gather*} \frac {20 \cdot 2^{\frac {1}{4}} x^{5} \arctan \left (2 \cdot 2^{\frac {3}{4}} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {1}{4}} {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x + \frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {2 \, x^{4} - 1} x^{2} + 2^{\frac {1}{4}} {\left (4 \, x^{4} - 1\right )}\right )}\right ) + 5 \cdot 2^{\frac {1}{4}} x^{5} \log \left (4 \, \sqrt {2} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {2 \, x^{4} - 1} x^{2} + 2^{\frac {3}{4}} {\left (4 \, x^{4} - 1\right )} + 4 \, {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x\right ) - 5 \cdot 2^{\frac {1}{4}} x^{5} \log \left (4 \, \sqrt {2} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {2 \, x^{4} - 1} x^{2} - 2^{\frac {3}{4}} {\left (4 \, x^{4} - 1\right )} + 4 \, {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x\right ) + 10 \, x^{5} \arctan \left (\frac {2 \, {\left ({\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - 1}\right ) + 10 \, x^{5} \log \left (-\frac {3 \, x^{4} - 2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {2 \, x^{4} - 1} x^{2} - 2 \, {\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{x^{4} - 1}\right ) + 8 \, {\left (2 \, x^{4} - 1\right )}^{\frac {5}{4}}}{40 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 143, normalized size = 1.35 \begin {gather*} -\frac {1}{2} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} - \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 2\right )}}{5 \, x} + \frac {1}{2} \, \arctan \left (\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) - \frac {1}{4} \, \log \left ({\left | \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 17.94, size = 448, normalized size = 4.23
method | result | size |
trager | \(\frac {\left (2 x^{4}-1\right )^{\frac {5}{4}}}{5 x^{5}}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{4}-2\right ) \sqrt {2 x^{4}-1}\, x^{2}+4 \left (2 x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}-2\right )^{3}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}-4 \sqrt {2 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+4 \left (2 x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )\right )}{4}-\frac {\ln \left (\frac {2 \left (2 x^{4}-1\right )^{\frac {3}{4}} x +2 \sqrt {2 x^{4}-1}\, x^{2}+2 \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}+3 x^{4}-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \sqrt {2 x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{4}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )+4 \left (2 x^{4}-1\right )^{\frac {3}{4}} x -4 \left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{8}\) | \(448\) |
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^{8} + x^{4} - 1\right )} {\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )} x^{6}}\,{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 (2\,x^4-1\right )}^{1/4}\,\left (x^8+x^4-1\right )}{x^6\,\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 {\sqrt [4]{2 x^{4} - 1} \left (x^{8} + x^{4} - 1\right )}{x^{6} \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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