Optimal. Leaf size=81 \[ \frac {x}{3 \sqrt [4]{x^4+1}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{6\ 2^{3/4} \sqrt [4]{3}} \]
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Rubi [A] time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1454, 527, 12, 377, 212, 206, 203} \begin {gather*} \frac {x}{3 \sqrt [4]{x^4+1}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{6\ 2^{3/4} \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 {1+2 x^4}{\sqrt [4]{1+x^4} \left (-2-x^4+x^8\right )} \, dx &=\int \frac {1+2 x^4}{\left (-2+x^4\right ) \left (1+x^4\right )^{5/4}} \, dx\\ &=\frac {x}{3 \sqrt [4]{1+x^4}}+\frac {1}{3} \int \frac {5}{\left (-2+x^4\right ) \sqrt [4]{1+x^4}} \, dx\\ &=\frac {x}{3 \sqrt [4]{1+x^4}}+\frac {5}{3} \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{1+x^4}} \, dx\\ &=\frac {x}{3 \sqrt [4]{1+x^4}}+\frac {5}{3} \operatorname {Subst}\left (\int \frac {1}{-2+3 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt {2}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{6 \sqrt {2}}\\ &=\frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 102, normalized size = 1.26 \begin {gather*} \frac {x}{3 \sqrt [4]{x^4+1}}-\frac {5 \left (-\log \left (2-\frac {2^{3/4} \sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac {2^{3/4} \sqrt [4]{3} x}{\sqrt [4]{x^4+1}}+2\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )\right )}{12\ 2^{3/4} \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 81, normalized size = 1.00 \begin {gather*} \frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.36, size = 245, normalized size = 3.02 \begin {gather*} \frac {20 \cdot 24^{\frac {3}{4}} {\left (x^{4} + 1\right )} \arctan \left (\frac {3 \cdot 24^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 12 \cdot 24^{\frac {1}{4}} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x + 6^{\frac {1}{4}} \sqrt {3} {\left (24^{\frac {3}{4}} \sqrt {x^{4} + 1} x^{2} + 24^{\frac {1}{4}} {\left (5 \, x^{4} + 2\right )}\right )}}{6 \, {\left (x^{4} - 2\right )}}\right ) - 5 \cdot 24^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 24 \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 24^{\frac {3}{4}} {\left (5 \, x^{4} + 2\right )} + 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) + 5 \cdot 24^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {24 \, \sqrt {6} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 24 \cdot 24^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 24^{\frac {3}{4}} {\left (5 \, x^{4} + 2\right )} + 48 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{4} - 2}\right ) + 192 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{576 \, {\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 {2 \, x^{4} + 1}{{\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 = 2.92, size = 222, normalized size = 2.74
method | result | size |
trager | \(\frac {x}{3 \left (x^{4}+1\right )^{\frac {1}{4}}}-\frac {5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x +6 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right )}{x^{4}-2}\right )}{72}-\frac {5 \RootOf \left (\textit {\_Z}^{4}-54\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-54\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} x^{3}+15 \RootOf \left (\textit {\_Z}^{4}-54\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x +6 \RootOf \left (\textit {\_Z}^{4}-54\right )}{x^{4}-2}\right )}{72}\) | \(222\) |
risch | \(\frac {x}{3 \left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {5 \RootOf \left (\textit {\_Z}^{4}-54\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-54\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \RootOf \left (\textit {\_Z}^{4}-54\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \RootOf \left (\textit {\_Z}^{4}-54\right )}{x^{4}-2}\right )}{72}-\frac {5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} x^{3}+15 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x +6 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right )}{x^{4}-2}\right )}{72}\) | \(222\) |
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 {2 \, x^{4} + 1}{{\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 {2\,x^4+1}{{\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 {2 x^{4} + 1}{\left (x^{4} - 2\right ) \left (x^{4} + 1\right )^{\frac {5}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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