Optimal. Leaf size=67 \[ \frac {1}{3} \tanh ^{-1}\left (\frac {(-1+x) \sqrt {1-4 x+3 x^2+2 x^3+x^4}}{x^3}\right )+\tanh ^{-1}\left (\frac {-1+2 x+\sqrt {1-4 x+3 x^2+2 x^3+x^4}}{x^2}\right ) \]
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Rubi [F]
time = 0.04, 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 {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx &=\int \frac {x}{\sqrt {1-4 x+3 x^2+2 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 11.16, size = 1189, normalized size = 17.75 \begin {gather*} \frac {\sqrt [4]{-1} \left (x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )^2 \sqrt {\frac {\left (-1+2 \sqrt [4]{-3}-i \sqrt {3}-2 x\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 x\right ) \left (-i+2 i \sqrt [4]{-3}+\sqrt {3}-2 i \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )^2}} \left (-2 F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(-1)^{3/4} \left (\frac {1}{2}+\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+x\right ) \left (\frac {1}{2}-\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )}{x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]}}}{\sqrt {2} \sqrt [8]{3}}\right )|\frac {2 \sqrt [4]{-3} \left (\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}{\left (\frac {1}{2}-\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (\frac {1}{2}+\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}\right ) \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]+\Pi \left (\frac {4 \sqrt [4]{-3}}{-1+2 \sqrt [4]{-3}-i \sqrt {3}-2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]};\text {ArcSin}\left (\frac {\sqrt {\frac {(-1)^{3/4} \left (\frac {1}{2}+\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+x\right ) \left (\frac {1}{2}-\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )}{x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]}}}{\sqrt {2} \sqrt [8]{3}}\right )|\frac {2 \sqrt [4]{-3} \left (\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}{\left (\frac {1}{2}-\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (\frac {1}{2}+\sqrt [4]{-3}+\frac {i \sqrt {3}}{2}+\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )\right ) \sqrt {\frac {\left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,3\right ]\right )}}}{\sqrt {1-4 x+3 x^2+2 x^3+x^4} \left (-1+2 \sqrt [4]{-3}-i \sqrt {3}-2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right ) \left (1+2 \sqrt [4]{-3}+i \sqrt {3}+2 \text {Root}\left [\text {$\#$1}^4+2 \text {$\#$1}^3+3 \text {$\#$1}^2-4 \text {$\#$1}+1\&,2\right ]\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 3.04, size = 1609, normalized size = 24.01
method | result | size |
trager | \(-\frac {\ln \left (-2 x^{6}+2 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{4}-12 x^{5}+10 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{3}-36 x^{4}+24 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}\, x^{2}-56 x^{3}+28 x \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}-42 x^{2}+14 \sqrt {x^{4}+2 x^{3}+3 x^{2}-4 x +1}+13\right )}{6}\) | \(151\) |
default | \(\text {Expression too large to display}\) | \(1609\) |
elliptic | \(\text {Expression too large to display}\) | \(1609\) |
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 [A]
time = 0.42, size = 70, normalized size = 1.04 \begin {gather*} \frac {1}{6} \, \log \left (2 \, x^{6} + 12 \, x^{5} + 36 \, x^{4} + 56 \, x^{3} + 42 \, x^{2} + 2 \, {\left (x^{4} + 5 \, x^{3} + 12 \, x^{2} + 14 \, x + 7\right )} \sqrt {x^{4} + 2 \, x^{3} + 3 \, x^{2} - 4 \, x + 1} - 13\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 {x}{\sqrt {x^{4} + 2 x^{3} + 3 x^{2} - 4 x + 1}}\, 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 {x}{\sqrt {x^4+2\,x^3+3\,x^2-4\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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