Optimal. Leaf size=105 \[ \frac {1}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}+\frac {1}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}} \]
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Rubi [A] time = 0.20, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6728, 377, 212, 206, 203} \begin {gather*} \frac {1}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}}+\frac {1}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 377
Rule 6728
Rubi steps
\begin {align*} \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx &=\int \left (-\frac {2}{3 \sqrt [4]{1+x^4} \left (-2+2 x^4\right )}+\frac {8}{3 \sqrt [4]{1+x^4} \left (4+2 x^4\right )}\right ) \, dx\\ &=-\left (\frac {2}{3} \int \frac {1}{\sqrt [4]{1+x^4} \left (-2+2 x^4\right )} \, dx\right )+\frac {8}{3} \int \frac {1}{\sqrt [4]{1+x^4} \left (4+2 x^4\right )} \, dx\\ &=-\left (\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-2+4 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\right )+\frac {8}{3} \operatorname {Subst}\left (\int \frac {1}{4-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{3} \sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{3} \sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {1}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 94, normalized size = 0.90 \begin {gather*} \frac {4 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )+4 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{6\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 105, normalized size = 1.00 \begin {gather*} \frac {1}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}}+\frac {1}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 9.74, size = 416, normalized size = 3.96 \begin {gather*} -\frac {1}{12} \cdot 8^{\frac {3}{4}} \arctan \left (-\frac {8^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 8^{\frac {1}{4}} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x - 2^{\frac {1}{4}} {\left (8^{\frac {3}{4}} \sqrt {x^{4} + 1} x^{2} + 8^{\frac {1}{4}} {\left (3 \, x^{4} + 2\right )}\right )}}{2 \, {\left (x^{4} + 2\right )}}\right ) - \frac {1}{12} \cdot 2^{\frac {3}{4}} \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 ) + \frac {1}{48} \cdot 8^{\frac {3}{4}} \log \left (\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 8^{\frac {3}{4}} {\left (3 \, x^{4} + 2\right )} + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) - \frac {1}{48} \cdot 8^{\frac {3}{4}} \log \left (\frac {8 \, \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 8^{\frac {3}{4}} {\left (3 \, x^{4} + 2\right )} + 16 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} + 2}\right ) + \frac {1}{48} \cdot 2^{\frac {3}{4}} \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 ) - \frac {1}{48} \cdot 2^{\frac {3}{4}} \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 ) \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*} \int \frac {x^{4} - 2}{{\left (x^{8} + x^{4} - 2\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.93, size = 457, normalized size = 4.35
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} \left (x^{4}+1\right )^{\frac {1}{4}} x^{3}+3 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-8\right ) \left (x^{4}+1\right )^{\frac {3}{4}} x +8 x^{2} \sqrt {x^{4}+1}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{24}+\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}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} 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 )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{24}+\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 {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{4}-4 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \sqrt {x^{4}+1}\, x^{2}+16 \left (x^{4}+1\right )^{\frac {3}{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^{4}+2}\right )}{24}+\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{4}+4 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+8 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{2}+16 \left (x^{4}+1\right )^{\frac {3}{4}} x +2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3}}{x^{4}+2}\right )}{24}\) | \(457\) |
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} - 2}{{\left (x^{8} + x^{4} - 2\right )} {\left (x^{4} + 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^4-2}{{\left (x^4+1\right )}^{1/4}\,\left (x^8+x^4-2\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 {x^{4} - 2}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt [4]{x^{4} + 1} \left (x^{4} + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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