Optimal. Leaf size=13 \[ \frac {8 \log (-1+x)}{4-x} \]
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Rubi [A] time = 0.10, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6742, 616, 31, 2395, 36} \begin {gather*} \frac {8 \log (x-1)}{4-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 616
Rule 2395
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {8}{4-5 x+x^2}+\frac {8 \log (-1+x)}{(-4+x)^2}\right ) \, dx\\ &=-\left (8 \int \frac {1}{4-5 x+x^2} \, dx\right )+8 \int \frac {\log (-1+x)}{(-4+x)^2} \, dx\\ &=\frac {8 \log (-1+x)}{4-x}-\frac {8}{3} \int \frac {1}{-4+x} \, dx+\frac {8}{3} \int \frac {1}{-1+x} \, dx+8 \int \frac {1}{(-4+x) (-1+x)} \, dx\\ &=\frac {8}{3} \log (1-x)-\frac {8}{3} \log (4-x)+\frac {8 \log (-1+x)}{4-x}+\frac {8}{3} \int \frac {1}{-4+x} \, dx-\frac {8}{3} \int \frac {1}{-1+x} \, dx\\ &=\frac {8 \log (-1+x)}{4-x}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.04, size = 42, normalized size = 3.23 \begin {gather*} \frac {8}{3} \left (2 \tanh ^{-1}\left (\frac {5}{3}-\frac {2 x}{3}\right )+\log (1-x)-\log (4-x)-\frac {3 \log (-1+x)}{-4+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 11, normalized size = 0.85 \begin {gather*} -\frac {8 \, \log \left (x - 1\right )}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 11, normalized size = 0.85 \begin {gather*} -\frac {8 \, \log \left (x - 1\right )}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 12, normalized size = 0.92
method | result | size |
norman | \(-\frac {8 \ln \left (x -1\right )}{x -4}\) | \(12\) |
risch | \(-\frac {8 \ln \left (x -1\right )}{x -4}\) | \(12\) |
derivativedivides | \(-\frac {8 \ln \left (x -1\right ) \left (x -1\right )}{3 \left (x -4\right )}+\frac {8 \ln \left (x -1\right )}{3}\) | \(22\) |
default | \(-\frac {8 \ln \left (x -1\right ) \left (x -1\right )}{3 \left (x -4\right )}+\frac {8 \ln \left (x -1\right )}{3}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 33, normalized size = 2.54 \begin {gather*} -\frac {8 \, {\left ({\left (4 \, x - 7\right )} \log \left (x - 1\right ) - 12\right )}}{9 \, {\left (x - 4\right )}} - \frac {32}{3 \, {\left (x - 4\right )}} + \frac {32}{9} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.69, size = 11, normalized size = 0.85 \begin {gather*} -\frac {8\,\ln \left (x-1\right )}{x-4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 10, normalized size = 0.77 \begin {gather*} - \frac {8 \log {\left (x - 1 \right )}}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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