Optimal. Leaf size=20 \[ x-\frac {x (2+\log (-4+x))}{(-3+x) \log (4)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(20)=40\).
time = 0.17, antiderivative size = 44, normalized size of antiderivative = 2.20, number of steps
used = 13, number of rules used = 8, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 6874, 46,
78, 90, 2442, 36, 31} \begin {gather*} x-\frac {\log (4-x)}{\log (4)}+\frac {3 \log (x-4)}{(3-x) \log (4)}+\frac {6}{(3-x) \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 36
Rule 46
Rule 78
Rule 90
Rule 2442
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-24+9 x-x^2+\left (-36+33 x-10 x^2+x^3\right ) \log (4)+(-12+3 x) \log (-4+x)}{-36+33 x-10 x^2+x^3} \, dx}{\log (4)}\\ &=\frac {\int \left (-\frac {24}{(-4+x) (-3+x)^2}+\frac {9 x}{(-4+x) (-3+x)^2}-\frac {x^2}{(-4+x) (-3+x)^2}+\log (4)+\frac {3 \log (-4+x)}{(-3+x)^2}\right ) \, dx}{\log (4)}\\ &=x-\frac {\int \frac {x^2}{(-4+x) (-3+x)^2} \, dx}{\log (4)}+\frac {3 \int \frac {\log (-4+x)}{(-3+x)^2} \, dx}{\log (4)}+\frac {9 \int \frac {x}{(-4+x) (-3+x)^2} \, dx}{\log (4)}-\frac {24 \int \frac {1}{(-4+x) (-3+x)^2} \, dx}{\log (4)}\\ &=x+\frac {3 \log (-4+x)}{(3-x) \log (4)}-\frac {\int \left (\frac {16}{-4+x}-\frac {9}{(-3+x)^2}-\frac {15}{-3+x}\right ) \, dx}{\log (4)}+\frac {3 \int \frac {1}{(-4+x) (-3+x)} \, dx}{\log (4)}+\frac {9 \int \left (\frac {4}{-4+x}-\frac {3}{(-3+x)^2}-\frac {4}{-3+x}\right ) \, dx}{\log (4)}-\frac {24 \int \left (\frac {1}{3-x}+\frac {1}{-4+x}-\frac {1}{(-3+x)^2}\right ) \, dx}{\log (4)}\\ &=x+\frac {6}{(3-x) \log (4)}+\frac {3 \log (3-x)}{\log (4)}-\frac {4 \log (4-x)}{\log (4)}+\frac {3 \log (-4+x)}{(3-x) \log (4)}+\frac {3 \int \frac {1}{-4+x} \, dx}{\log (4)}-\frac {3 \int \frac {1}{-3+x} \, dx}{\log (4)}\\ &=x+\frac {6}{(3-x) \log (4)}-\frac {\log (4-x)}{\log (4)}+\frac {3 \log (-4+x)}{(3-x) \log (4)}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(20)=40\).
time = 0.10, size = 49, normalized size = 2.45 \begin {gather*} \frac {6 \tanh ^{-1}(7-2 x)+x \log (4)+3 \log (3-x)-4 \log (4-x)+\frac {3 (2+\log (-4+x))}{3-x}}{\log (4)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs.
\(2(20)=40\).
time = 0.75, size = 42, normalized size = 2.10
method | result | size |
norman | \(\frac {x^{2}-\frac {3 \left (3 \ln \left (2\right )+1\right )}{\ln \left (2\right )}-\frac {x \ln \left (x -4\right )}{2 \ln \left (2\right )}}{x -3}\) | \(34\) |
derivativedivides | \(\frac {\frac {3 \ln \left (x -4\right ) \left (x -4\right )}{x -3}-\frac {6}{x -3}-4 \ln \left (x -4\right )+2 \left (x -4\right ) \ln \left (2\right )}{2 \ln \left (2\right )}\) | \(42\) |
default | \(\frac {\frac {3 \ln \left (x -4\right ) \left (x -4\right )}{x -3}-\frac {6}{x -3}-4 \ln \left (x -4\right )+2 \left (x -4\right ) \ln \left (2\right )}{2 \ln \left (2\right )}\) | \(42\) |
risch | \(-\frac {3 \ln \left (x -4\right )}{2 \ln \left (2\right ) \left (x -3\right )}+\frac {2 x^{2} \ln \left (2\right )-6 x \ln \left (2\right )-x \ln \left (x -4\right )+3 \ln \left (x -4\right )-6}{2 \ln \left (2\right ) \left (x -3\right )}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (20) = 40\).
time = 0.49, size = 102, normalized size = 5.10 \begin {gather*} -\frac {72 \, {\left (\frac {1}{x - 3} - \log \left (x - 3\right ) + \log \left (x - 4\right )\right )} \log \left (2\right ) + 24 \, {\left (3 \, \log \left (2\right ) + 1\right )} \log \left (x - 3\right ) - \frac {2 \, x^{2} \log \left (2\right ) - 6 \, x \log \left (2\right ) + {\left (x {\left (72 \, \log \left (2\right ) + 23\right )} - 216 \, \log \left (2\right ) - 72\right )} \log \left (x - 4\right ) + 72 \, \log \left (2\right ) + 18}{x - 3} + \frac {24}{x - 3} - 24 \, \log \left (x - 3\right ) + 24 \, \log \left (x - 4\right )}{2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 31, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (x^{2} - 3 \, x\right )} \log \left (2\right ) - x \log \left (x - 4\right ) - 6}{2 \, {\left (x - 3\right )} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (17) = 34\).
time = 0.21, size = 41, normalized size = 2.05 \begin {gather*} x - \frac {\log {\left (x - 4 \right )}}{2 \log {\left (2 \right )}} - \frac {3 \log {\left (x - 4 \right )}}{2 x \log {\left (2 \right )} - 6 \log {\left (2 \right )}} - \frac {3}{x \log {\left (2 \right )} - 3 \log {\left (2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 36, normalized size = 1.80 \begin {gather*} \frac {2 \, x \log \left (2\right ) - \frac {3 \, \log \left (x - 4\right )}{x - 3} - \frac {6}{x - 3} - \log \left (x - 4\right )}{2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 36, normalized size = 1.80 \begin {gather*} -\frac {\ln \left (\frac {x-4}{2^{2\,x}}\right )}{2\,\ln \left (2\right )}-\frac {3\,\left (\ln \left (x-4\right )+2\right )}{2\,\ln \left (2\right )\,\left (x-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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