Optimal. Leaf size=25 \[ \frac {\left (1+x+\left (2+\frac {1}{x^2}\right ) x\right ) \left (3+x^2\right ) \log (x)}{(1-x)^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(25)=50\).
time = 0.54, antiderivative size = 74, normalized size of antiderivative = 2.96, number of steps
used = 19, number of rules used = 11, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.153, Rules used = {6873, 6874,
46, 37, 45, 2404, 2332, 2356, 2351, 31, 2341} \begin {gather*} -\frac {9 x^2}{2 (1-x)^2}-\frac {9}{1-x}+\frac {9}{2 (1-x)^2}-\frac {18 x \log (x)}{1-x}+3 x \log (x)+\frac {20 \log (x)}{(1-x)^2}-11 \log (x)+\frac {3 \log (x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 37
Rule 45
Rule 46
Rule 2332
Rule 2341
Rule 2351
Rule 2356
Rule 2404
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+7 x^2-9 x^3+2 x^4-3 x^5-\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{(1-x)^3 x^2} \, dx\\ &=\int \left (-\frac {7}{(-1+x)^3}-\frac {3}{(-1+x)^3 x^2}+\frac {9 x}{(-1+x)^3}-\frac {2 x^2}{(-1+x)^3}+\frac {3 x^3}{(-1+x)^3}+\frac {\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{(-1+x)^3 x^2}\right ) \, dx\\ &=\frac {7}{2 (1-x)^2}-2 \int \frac {x^2}{(-1+x)^3} \, dx-3 \int \frac {1}{(-1+x)^3 x^2} \, dx+3 \int \frac {x^3}{(-1+x)^3} \, dx+9 \int \frac {x}{(-1+x)^3} \, dx+\int \frac {\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{(-1+x)^3 x^2} \, dx\\ &=\frac {7}{2 (1-x)^2}-\frac {9 x^2}{2 (1-x)^2}-2 \int \left (\frac {1}{(-1+x)^3}+\frac {2}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx+3 \int \left (1+\frac {1}{(-1+x)^3}+\frac {3}{(-1+x)^2}+\frac {3}{-1+x}\right ) \, dx-3 \int \left (\frac {1}{(-1+x)^3}-\frac {2}{(-1+x)^2}+\frac {3}{-1+x}-\frac {1}{x^2}-\frac {3}{x}\right ) \, dx+\int \left (3 \log (x)-\frac {40 \log (x)}{(-1+x)^3}-\frac {18 \log (x)}{(-1+x)^2}-\frac {3 \log (x)}{x^2}\right ) \, dx\\ &=\frac {9}{2 (1-x)^2}+\frac {11}{1-x}-\frac {3}{x}+3 x-\frac {9 x^2}{2 (1-x)^2}-2 \log (1-x)+9 \log (x)+3 \int \log (x) \, dx-3 \int \frac {\log (x)}{x^2} \, dx-18 \int \frac {\log (x)}{(-1+x)^2} \, dx-40 \int \frac {\log (x)}{(-1+x)^3} \, dx\\ &=\frac {9}{2 (1-x)^2}+\frac {11}{1-x}-\frac {9 x^2}{2 (1-x)^2}-2 \log (1-x)+9 \log (x)+\frac {20 \log (x)}{(1-x)^2}+\frac {3 \log (x)}{x}+3 x \log (x)-\frac {18 x \log (x)}{1-x}-18 \int \frac {1}{-1+x} \, dx-20 \int \frac {1}{(-1+x)^2 x} \, dx\\ &=\frac {9}{2 (1-x)^2}+\frac {11}{1-x}-\frac {9 x^2}{2 (1-x)^2}-20 \log (1-x)+9 \log (x)+\frac {20 \log (x)}{(1-x)^2}+\frac {3 \log (x)}{x}+3 x \log (x)-\frac {18 x \log (x)}{1-x}-20 \int \left (\frac {1}{1-x}+\frac {1}{(-1+x)^2}+\frac {1}{x}\right ) \, dx\\ &=\frac {9}{2 (1-x)^2}-\frac {9}{1-x}-\frac {9 x^2}{2 (1-x)^2}-11 \log (x)+\frac {20 \log (x)}{(1-x)^2}+\frac {3 \log (x)}{x}+3 x \log (x)-\frac {18 x \log (x)}{1-x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 29, normalized size = 1.16 \begin {gather*} \frac {\left (3+3 x+10 x^2+x^3+3 x^4\right ) \log (x)}{(-1+x)^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 41, normalized size = 1.64
method | result | size |
norman | \(\frac {6 x^{3} \ln \left (x \right )+8 x \ln \left (x \right )+3 x^{4} \ln \left (x \right )+3 \ln \left (x \right )}{x \left (x -1\right )^{2}}-5 \ln \left (x \right )\) | \(39\) |
default | \(9 \ln \left (x \right )+3 x \ln \left (x \right )+\frac {3 \ln \left (x \right )}{x}+\frac {18 \ln \left (x \right ) x}{x -1}-\frac {20 \ln \left (x \right ) x \left (x -2\right )}{\left (x -1\right )^{2}}\) | \(41\) |
risch | \(\frac {\left (3 x^{4}-6 x^{3}+24 x^{2}-4 x +3\right ) \ln \left (x \right )}{x \left (x^{2}-2 x +1\right )}+7 \ln \left (x \right )\) | \(42\) |
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. 152 vs.
\(2 (23) = 46\).
time = 0.30, size = 152, normalized size = 6.08 \begin {gather*} 3 \, x - \frac {3 \, x^{4} - 6 \, x^{3} - 20 \, x^{2} - {\left (3 \, x^{4} - 6 \, x^{3} + 24 \, x^{2} - 4 \, x + 3\right )} \log \left (x\right ) + 26 \, x - 3}{x^{3} - 2 \, x^{2} + x} - \frac {3 \, {\left (6 \, x^{2} - 9 \, x + 2\right )}}{2 \, {\left (x^{3} - 2 \, x^{2} + x\right )}} - \frac {3 \, {\left (6 \, x - 5\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {4 \, x - 3}{x^{2} - 2 \, x + 1} - \frac {9 \, {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {7}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + 7 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 33, normalized size = 1.32 \begin {gather*} \frac {{\left (3 \, x^{4} + x^{3} + 10 \, x^{2} + 3 \, x + 3\right )} \log \left (x\right )}{x^{3} - 2 \, x^{2} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 37, normalized size = 1.48 \begin {gather*} 7 \log {\left (x \right )} + \frac {\left (3 x^{4} - 6 x^{3} + 24 x^{2} - 4 x + 3\right ) \log {\left (x \right )}}{x^{3} - 2 x^{2} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 34, normalized size = 1.36 \begin {gather*} {\left (3 \, x + \frac {2 \, {\left (9 \, x + 1\right )}}{x^{2} - 2 \, x + 1} + \frac {3}{x}\right )} \log \left (x\right ) + 7 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.89, size = 29, normalized size = 1.16 \begin {gather*} \frac {\ln \left (x\right )\,\left (3\,x^4+x^3+10\,x^2+3\,x+3\right )}{x\,{\left (x-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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