Optimal. Leaf size=21 \[ \frac {x}{4}-\frac {45 x^2}{4}+\log (-2+x)+\log (2 x) \]
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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1607, 1634}
\begin {gather*} -\frac {45 x^2}{4}+\frac {x}{4}+\log (2-x)+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 1607
Rule 1634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8+6 x+181 x^2-90 x^3}{x (-8+4 x)} \, dx\\ &=\int \left (\frac {1}{4}+\frac {1}{-2+x}+\frac {1}{x}-\frac {45 x}{2}\right ) \, dx\\ &=\frac {x}{4}-\frac {45 x^2}{4}+\log (2-x)+\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {x}{4}-\frac {45 x^2}{4}+\log (2-x)+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.17, size = 16, normalized size = 0.76
method | result | size |
default | \(-\frac {45 x^{2}}{4}+\frac {x}{4}+\ln \left (x -2\right )+\ln \left (x \right )\) | \(16\) |
norman | \(-\frac {45 x^{2}}{4}+\frac {x}{4}+\ln \left (x -2\right )+\ln \left (x \right )\) | \(16\) |
risch | \(\frac {x}{4}-\frac {45 x^{2}}{4}+\ln \left (x^{2}-2 x \right )\) | \(18\) |
meijerg | \(\ln \left (1-\frac {x}{2}\right )+\ln \left (x \right )-\ln \left (2\right )+i \pi -\frac {15 x \left (\frac {3 x}{2}+6\right )}{2}+\frac {181 x}{4}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 15, normalized size = 0.71 \begin {gather*} -\frac {45}{4} \, x^{2} + \frac {1}{4} \, x + \log \left (x - 2\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 17, normalized size = 0.81 \begin {gather*} -\frac {45}{4} \, x^{2} + \frac {1}{4} \, x + \log \left (x^{2} - 2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 17, normalized size = 0.81 \begin {gather*} - \frac {45 x^{2}}{4} + \frac {x}{4} + \log {\left (x^{2} - 2 x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 17, normalized size = 0.81 \begin {gather*} -\frac {45}{4} \, x^{2} + \frac {1}{4} \, x + \log \left ({\left | x - 2 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.08, size = 15, normalized size = 0.71 \begin {gather*} \frac {x}{4}+\ln \left (x\,\left (x-2\right )\right )-\frac {45\,x^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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