Optimal. Leaf size=21 \[ 10+\frac {x^2 (x+\log (x))}{\left (x-\frac {x^2}{4}\right )^2} \]
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Rubi [A]
time = 0.15, antiderivative size = 33, normalized size of antiderivative = 1.57, number of steps
used = 10, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6820, 12,
6874, 46, 37, 2356} \begin {gather*} \frac {2 x^2}{(4-x)^2}+\frac {32}{(4-x)^2}+\frac {16 \log (x)}{(4-x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 46
Rule 2356
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 \left (4+3 x+x^2+2 x \log (x)\right )}{(4-x)^3 x} \, dx\\ &=16 \int \frac {4+3 x+x^2+2 x \log (x)}{(4-x)^3 x} \, dx\\ &=16 \int \left (-\frac {3}{(-4+x)^3}-\frac {4}{(-4+x)^3 x}-\frac {x}{(-4+x)^3}-\frac {2 \log (x)}{(-4+x)^3}\right ) \, dx\\ &=\frac {24}{(4-x)^2}-16 \int \frac {x}{(-4+x)^3} \, dx-32 \int \frac {\log (x)}{(-4+x)^3} \, dx-64 \int \frac {1}{(-4+x)^3 x} \, dx\\ &=\frac {24}{(4-x)^2}+\frac {2 x^2}{(4-x)^2}+\frac {16 \log (x)}{(4-x)^2}-16 \int \frac {1}{(-4+x)^2 x} \, dx-64 \int \left (\frac {1}{4 (-4+x)^3}-\frac {1}{16 (-4+x)^2}+\frac {1}{64 (-4+x)}-\frac {1}{64 x}\right ) \, dx\\ &=\frac {32}{(4-x)^2}+\frac {4}{4-x}+\frac {2 x^2}{(4-x)^2}-\log (4-x)+\log (x)+\frac {16 \log (x)}{(4-x)^2}-16 \int \left (\frac {1}{4 (-4+x)^2}-\frac {1}{16 (-4+x)}+\frac {1}{16 x}\right ) \, dx\\ &=\frac {32}{(4-x)^2}+\frac {2 x^2}{(4-x)^2}+\frac {16 \log (x)}{(4-x)^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 11, normalized size = 0.52 \begin {gather*} \frac {16 (x+\log (x))}{(-4+x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 31, normalized size = 1.48
method | result | size |
norman | \(\frac {16 x +16 \ln \left (x \right )}{\left (x -4\right )^{2}}\) | \(15\) |
risch | \(\frac {16 \ln \left (x \right )}{x^{2}-8 x +16}+\frac {16 x}{x^{2}-8 x +16}\) | \(29\) |
default | \(\frac {16}{x -4}-\frac {\ln \left (x \right ) x \left (-8+x \right )}{\left (x -4\right )^{2}}+\ln \left (x \right )+\frac {64}{\left (x -4\right )^{2}}\) | \(31\) |
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. 64 vs.
\(2 (20) = 40\).
time = 0.26, size = 64, normalized size = 3.05 \begin {gather*} \frac {16 \, {\left (x - 2\right )}}{x^{2} - 8 \, x + 16} - \frac {4 \, {\left (x - 6\right )}}{x^{2} - 8 \, x + 16} + \frac {16 \, \log \left (x\right )}{x^{2} - 8 \, x + 16} + \frac {24}{x^{2} - 8 \, x + 16} + \frac {4}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 16, normalized size = 0.76 \begin {gather*} \frac {16 \, {\left (x + \log \left (x\right )\right )}}{x^{2} - 8 \, x + 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 24, normalized size = 1.14 \begin {gather*} \frac {16 x}{x^{2} - 8 x + 16} + \frac {16 \log {\left (x \right )}}{x^{2} - 8 x + 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 28, normalized size = 1.33 \begin {gather*} \frac {16 \, x}{x^{2} - 8 \, x + 16} + \frac {16 \, \log \left (x\right )}{x^{2} - 8 \, x + 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.13, size = 11, normalized size = 0.52 \begin {gather*} \frac {16\,\left (x+\ln \left (x\right )\right )}{{\left (x-4\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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