Optimal. Leaf size=64 \[ \frac {x}{\sqrt [4]{x^4+1}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1454, 527, 12, 377, 212, 206, 203} \begin {gather*} \frac {x}{\sqrt [4]{x^4+1}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 206
Rule 212
Rule 377
Rule 527
Rule 1454
Rubi steps
\begin {align*} \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx &=\int \frac {-2+x^4}{\left (1+x^4\right )^{5/4} \left (-1+2 x^4\right )} \, dx\\ &=\frac {x}{\sqrt [4]{1+x^4}}+\frac {1}{3} \int -\frac {3}{\sqrt [4]{1+x^4} \left (-1+2 x^4\right )} \, dx\\ &=\frac {x}{\sqrt [4]{1+x^4}}-\int \frac {1}{\sqrt [4]{1+x^4} \left (-1+2 x^4\right )} \, dx\\ &=\frac {x}{\sqrt [4]{1+x^4}}-\operatorname {Subst}\left (\int \frac {1}{-1+3 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {x}{\sqrt [4]{1+x^4}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {x}{\sqrt [4]{1+x^4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 82, normalized size = 1.28 \begin {gather*} \frac {x}{\sqrt [4]{x^4+1}}+\frac {-\log \left (1-\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}+1\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.33, size = 64, normalized size = 1.00 \begin {gather*} \frac {x}{\sqrt [4]{1+x^4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.36, size = 248, normalized size = 3.88 \begin {gather*} -\frac {4 \cdot 3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \arctan \left (\frac {6 \cdot 3^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 3^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {x^{4} + 1} x^{2} + 3^{\frac {1}{4}} {\left (4 \, x^{4} + 1\right )}\right )}}{3 \, {\left (2 \, x^{4} - 1\right )}}\right ) - 3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 3^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) + 3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 3^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) - 24 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{24 \, {\left (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 (2 \, x^{8} + x^{4} - 1\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 = 2.74, size = 225, normalized size = 3.52
method | result | size |
trager | \(\frac {x}{\left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {\RootOf \left (\textit {\_Z}^{4}-27\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-27\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-27\right )^{2} x^{3}+12 \RootOf \left (\textit {\_Z}^{4}-27\right ) x^{4}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x +3 \RootOf \left (\textit {\_Z}^{4}-27\right )}{2 x^{4}-1}\right )}{12}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-27\right )^{2}\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-27\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-27\right )^{2} x^{3}+12 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{4}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x +3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-27\right )^{2}\right )}{2 x^{4}-1}\right )}{12}\) | \(225\) |
risch | \(\frac {x}{\left (x^{4}+1\right )^{\frac {1}{4}}}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-27\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-27\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-27\right )^{2} x^{3}-12 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{4}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x -3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-27\right )^{2}\right )}{2 x^{4}-1}\right )}{12}+\frac {\RootOf \left (\textit {\_Z}^{4}-27\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-27\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-27\right )^{2} x^{3}+12 \RootOf \left (\textit {\_Z}^{4}-27\right ) x^{4}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x +3 \RootOf \left (\textit {\_Z}^{4}-27\right )}{2 x^{4}-1}\right )}{12}\) | \(225\) |
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 (2 \, x^{8} + x^{4} - 1\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 = -0.02 \begin {gather*} \int \frac {x^4-2}{{\left (x^4+1\right )}^{1/4}\,\left (2\,x^8+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 {x^{4} - 2}{\left (x^{4} + 1\right )^{\frac {5}{4}} \left (2 x^{4} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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