Optimal. Leaf size=23 \[ -x+x \left (2 x^2+\log \left (\frac {1}{2} x (-1+4 x)\right )\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 22, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {6742, 771, 2487, 31, 8} \begin {gather*} 2 x^3-x+x \log \left (-\frac {1}{2} (1-4 x) x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 31
Rule 771
Rule 2487
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 x \left (2-3 x+12 x^2\right )}{-1+4 x}+\log \left (\frac {1}{2} x (-1+4 x)\right )\right ) \, dx\\ &=2 \int \frac {x \left (2-3 x+12 x^2\right )}{-1+4 x} \, dx+\int \log \left (\frac {1}{2} x (-1+4 x)\right ) \, dx\\ &=x \log \left (-\frac {1}{2} (1-4 x) x\right )-2 \int 1 \, dx+2 \int \left (\frac {1}{2}+3 x^2+\frac {1}{2 (-1+4 x)}\right ) \, dx-\int \frac {1}{-1+4 x} \, dx\\ &=-x+2 x^3+x \log \left (-\frac {1}{2} (1-4 x) x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.96 \begin {gather*} -\frac {9}{32}-x+2 x^3-\frac {1}{4} \log (1-4 x)+\frac {1}{4} \log (-1+4 x)+x \log \left (\frac {1}{2} x (-1+4 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 21, normalized size = 0.91 \begin {gather*} 2 \, x^{3} + x \log \left (2 \, x^{2} - \frac {1}{2} \, x\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 21, normalized size = 0.91 \begin {gather*} 2 \, x^{3} + x \log \left (2 \, x^{2} - \frac {1}{2} \, x\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 22, normalized size = 0.96
method | result | size |
norman | \(\ln \left (2 x^{2}-\frac {1}{2} x \right ) x -x +2 x^{3}\) | \(22\) |
risch | \(\ln \left (2 x^{2}-\frac {1}{2} x \right ) x -x +2 x^{3}\) | \(22\) |
default | \(2 x^{3}-x -x \ln \relax (2)+x \ln \left (4 x^{2}-x \right )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 39, normalized size = 1.70 \begin {gather*} 2 \, x^{3} - x {\left (\log \relax (2) + 2\right )} + \frac {1}{4} \, {\left (4 \, x - 1\right )} \log \left (4 \, x - 1\right ) + x \log \relax (x) + x + \frac {1}{4} \, \log \left (4 \, x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 20, normalized size = 0.87 \begin {gather*} x\,\left (\ln \left (2\,x^2-\frac {x}{2}\right )-1\right )+2\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 17, normalized size = 0.74 \begin {gather*} 2 x^{3} + x \log {\left (2 x^{2} - \frac {x}{2} \right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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