Optimal. Leaf size=86 \[ -\frac {\left (x^4-x^2\right )^{3/4}}{x \left (x^2-1\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^2}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^2}}\right )}{2 \sqrt [4]{2}} \]
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Rubi [A] time = 0.12, antiderivative size = 131, normalized size of antiderivative = 1.52, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2056, 1254, 466, 382, 377, 212, 206, 203} \begin {gather*} -\frac {x}{\sqrt [4]{x^4-x^2}}-\frac {\sqrt [4]{x^2-1} \sqrt {x} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4-x^2}}-\frac {\sqrt [4]{x^2-1} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{2 \sqrt [4]{2} \sqrt [4]{x^4-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 382
Rule 466
Rule 1254
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-1+x^2} \left (-1+x^4\right )} \, dx}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1+x^2\right )^{5/4} \left (1+x^2\right )} \, dx}{\sqrt [4]{-x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^4\right )^{5/4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=-\frac {x}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=-\frac {x}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt [4]{-x^2+x^4}}\\ &=-\frac {x}{\sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{-x^2+x^4}}\\ &=-\frac {x}{\sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}-\frac {\sqrt {x} \sqrt [4]{-1+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt [4]{2} \sqrt [4]{-x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 69, normalized size = 0.80 \begin {gather*} -\frac {x \left (\sqrt [4]{1-x^2} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^2}{x^2+1}\right )+\sqrt [4]{x^2+1}\right )}{\sqrt [4]{x^2 \left (x^2-1\right )} \sqrt [4]{x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.26, size = 86, normalized size = 1.00 \begin {gather*} -\frac {\left (x^4-x^2\right )^{3/4}}{x \left (x^2-1\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^2}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-x^2}}\right )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.65, size = 286, normalized size = 3.33 \begin {gather*} \frac {4 \cdot 2^{\frac {3}{4}} {\left (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 ) - 2^{\frac {3}{4}} {\left (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 ) + 2^{\frac {3}{4}} {\left (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 (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{16 \, {\left (x^{3} - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 70, normalized size = 0.81 \begin {gather*} -\frac {1}{4} \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}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {1}{4}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.02, size = 247, normalized size = 2.87 \begin {gather*} -\frac {x}{\left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}-\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 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}} 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 (x^{2}+1\right )}\right )}{8}+\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 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}} 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}{x \left (x^{2}+1\right )}\right )}{8} \end {gather*}
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 {1}{{\left (x^{4} - 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 {1}{\left (x^4-1\right )\,{\left (x^4-x^2\right )}^{1/4}} \,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 {1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 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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