Optimal. Leaf size=81 \[ \frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \text {ArcTan}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{1+x^4}}\right )}{6\ 2^{3/4} \sqrt [4]{3}} \]
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Rubi [A]
time = 0.05, 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 = {1468, 541, 12,
385, 218, 212, 209} \begin {gather*} -\frac {5 \text {ArcTan}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{x^4+1}}\right )}{6\ 2^{3/4} \sqrt [4]{3}}+\frac {x}{3 \sqrt [4]{x^4+1}}-\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 209
Rule 212
Rule 218
Rule 385
Rule 541
Rule 1468
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} \text {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 \text {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 \text {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 = 0.29, size = 81, normalized size = 1.00 \begin {gather*} \frac {x}{3 \sqrt [4]{1+x^4}}-\frac {5 \text {ArcTan}\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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.82, 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 | \(\frac {x}{3 \left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-54\right )^{2}\right )}{x^{4}-2}\right )}{72}+\frac {5 \RootOf \left (\textit {\_Z}^{4}-54\right ) \ln \left (-\frac {-2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}-54\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-54\right )^{2} x^{3}-15 \RootOf \left (\textit {\_Z}^{4}-54\right ) x^{4}+36 \left (x^{4}+1\right )^{\frac {3}{4}} x -6 \RootOf \left (\textit {\_Z}^{4}-54\right )}{x^{4}-2}\right )}{72}\) | \(222\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs.
\(2 (61) = 122\).
time = 3.30, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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