Optimal. Leaf size=78 \[ -\frac {4 \sqrt [4]{-2-x+2 x^4}}{x}-2 \sqrt [4]{3} \text {ArcTan}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right )+2 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right ) \]
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Rubi [F]
time = 0.64, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx &=\int \left (\frac {4 \sqrt [4]{-2-x+2 x^4}}{x^2}-\frac {\sqrt [4]{-2-x+2 x^4}}{2 x}+\frac {\left (1-8 x^2+x^3\right ) \sqrt [4]{-2-x+2 x^4}}{2 \left (2+x+x^4\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt [4]{-2-x+2 x^4}}{x} \, dx\right )+\frac {1}{2} \int \frac {\left (1-8 x^2+x^3\right ) \sqrt [4]{-2-x+2 x^4}}{2+x+x^4} \, dx+4 \int \frac {\sqrt [4]{-2-x+2 x^4}}{x^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt [4]{-2-x+2 x^4}}{x} \, dx\right )+\frac {1}{2} \int \left (\frac {\sqrt [4]{-2-x+2 x^4}}{2+x+x^4}-\frac {8 x^2 \sqrt [4]{-2-x+2 x^4}}{2+x+x^4}+\frac {x^3 \sqrt [4]{-2-x+2 x^4}}{2+x+x^4}\right ) \, dx+4 \int \frac {\sqrt [4]{-2-x+2 x^4}}{x^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt [4]{-2-x+2 x^4}}{x} \, dx\right )+\frac {1}{2} \int \frac {\sqrt [4]{-2-x+2 x^4}}{2+x+x^4} \, dx+\frac {1}{2} \int \frac {x^3 \sqrt [4]{-2-x+2 x^4}}{2+x+x^4} \, dx+4 \int \frac {\sqrt [4]{-2-x+2 x^4}}{x^2} \, dx-4 \int \frac {x^2 \sqrt [4]{-2-x+2 x^4}}{2+x+x^4} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 78, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{-2-x+2 x^4}}{x}-2 \sqrt [4]{3} \text {ArcTan}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right )+2 \sqrt [4]{3} \tanh ^{-1}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right ) \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 = 8.03, size = 303, normalized size = 3.88
method | result | size |
trager | \(-\frac {4 \left (2 x^{4}-x -2\right )^{\frac {1}{4}}}{x}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{4}-6 \left (2 x^{4}-x -2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{3}-6 \sqrt {2 x^{4}-x -2}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{2}+6 \left (2 x^{4}-x -2\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right ) x -2 \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-3\right )^{2}\right )}{x^{4}+x +2}\right )+\RootOf \left (\textit {\_Z}^{4}-3\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x^{4}+6 \left (2 x^{4}-x -2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-3\right )^{2} x^{3}+6 \sqrt {2 x^{4}-x -2}\, \RootOf \left (\textit {\_Z}^{4}-3\right ) x^{2}+6 \left (2 x^{4}-x -2\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}-3\right )^{3} x -2 \RootOf \left (\textit {\_Z}^{4}-3\right )^{3}}{x^{4}+x +2}\right )\) | \(303\) |
risch | \(\text {Expression too large to display}\) | \(1594\) |
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. 287 vs.
\(2 (64) = 128\).
time = 6.73, size = 287, normalized size = 3.68 \begin {gather*} \frac {4 \cdot 3^{\frac {1}{4}} x \arctan \left (\frac {6 \cdot 3^{\frac {3}{4}} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x + 3^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} + 3^{\frac {1}{4}} {\left (5 \, x^{4} - x - 2\right )}\right )}}{3 \, {\left (x^{4} + x + 2\right )}}\right ) + 3^{\frac {1}{4}} x \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} + 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 2\right )} + 6 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x}{x^{4} + x + 2}\right ) - 3^{\frac {1}{4}} x \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} - 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 2\right )} + 6 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x}{x^{4} + x + 2}\right ) - 8 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (3\,x+8\right )\,{\left (2\,x^4-x-2\right )}^{1/4}}{x^2\,\left (x^4+x+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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