Optimal. Leaf size=34 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {x^2+x-2}{\sqrt {x^4+2 x^3-3 x^2+3 x-3}}\right ) \]
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Rubi [F] time = 0.07, 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 {-3+3 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 {-3+3 x-3 x^2+2 x^3+x^4}} \, dx &=\int \frac {x}{\sqrt {-3+3 x-3 x^2+2 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.93, size = 768, normalized size = 22.59 \begin {gather*} \frac {2 (x-1) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-1\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]\right )}} \left (\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-1\right ) \Pi \left (\frac {-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]};\sin ^{-1}\left (\sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}\right )-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ] F\left (\sin ^{-1}\left (\sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}\right )\right )}{\sqrt {x^4+2 x^3-3 x^2+3 x-3} \left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right ) \sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-1\right ) \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,3\right ]\right )^2 \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+3\&,1\right ]\right )^2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.16, size = 34, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {x^2+x-2}{\sqrt {x^4+2 x^3-3 x^2+3 x-3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 40, normalized size = 1.18 \begin {gather*} \frac {1}{3} \, \log \left (2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} + 3 \, x - 3} {\left (x + 2\right )} - 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}{\sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} + 3 \, x - 3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.25, size = 1633, normalized size = 48.03 \begin {gather*} \text {Expression too large to display} \end {gather*}
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}{\sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} + 3 \, x - 3}}\,{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 = -0.03 \begin {gather*} \int \frac {x}{\sqrt {x^4+2\,x^3-3\,x^2+3\,x-3}} \,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}{\sqrt {\left (x - 1\right ) \left (x^{3} + 3 x^{2} + 3\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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