Optimal. Leaf size=31 \[ -x+\log (x)+\frac {\left (-5+e^{5-x}\right ) x^2 (4+\log (x))}{3 (-3+x)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(159\) vs. \(2(31)=62\).
time = 1.35, antiderivative size = 159, normalized size of antiderivative = 5.13, number of steps
used = 35, number of rules used = 17, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.181, Rules used = {1608, 27,
12, 6874, 46, 45, 2384, 2353, 2352, 2393, 2332, 2230, 2225, 2208, 2209, 2207, 2634}
\begin {gather*} -\frac {5 x^2 \log (x)}{3 (3-x)}+\frac {4}{3} e^{5-x} x-\frac {23 x}{3}+4 e^{5-x}-\frac {12 e^{5-x}}{3-x}+\frac {60}{3-x}+\frac {1}{3} e^{5-x} x \log (x)+\frac {10 x \log (x)}{3-x}-\frac {10}{3} x \log (x)-5 \log (3-x)-5 (1+\log (9)) \log (x-3)+10 (1+\log (3)) \log (x-3)+e^{5-x} \log (x)-\frac {3 e^{5-x} \log (x)}{3-x}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 45
Rule 46
Rule 1608
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 2332
Rule 2352
Rule 2353
Rule 2384
Rule 2393
Rule 2634
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27-45 x+156 x^2-28 x^3+e^{5-x} \left (-27 x^2+17 x^3-4 x^4\right )+\left (30 x^2-5 x^3+e^{5-x} \left (-6 x^2+4 x^3-x^4\right )\right ) \log (x)}{x \left (27-18 x+3 x^2\right )} \, dx\\ &=\int \frac {27-45 x+156 x^2-28 x^3+e^{5-x} \left (-27 x^2+17 x^3-4 x^4\right )+\left (30 x^2-5 x^3+e^{5-x} \left (-6 x^2+4 x^3-x^4\right )\right ) \log (x)}{3 (-3+x)^2 x} \, dx\\ &=\frac {1}{3} \int \frac {27-45 x+156 x^2-28 x^3+e^{5-x} \left (-27 x^2+17 x^3-4 x^4\right )+\left (30 x^2-5 x^3+e^{5-x} \left (-6 x^2+4 x^3-x^4\right )\right ) \log (x)}{(-3+x)^2 x} \, dx\\ &=\frac {1}{3} \int \left (-\frac {45}{(-3+x)^2}+\frac {27}{(-3+x)^2 x}+\frac {156 x}{(-3+x)^2}-\frac {28 x^2}{(-3+x)^2}+\frac {30 x \log (x)}{(-3+x)^2}-\frac {5 x^2 \log (x)}{(-3+x)^2}-\frac {e^{5-x} x \left (27-17 x+4 x^2+6 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{(-3+x)^2}\right ) \, dx\\ &=-\frac {15}{3-x}-\frac {1}{3} \int \frac {e^{5-x} x \left (27-17 x+4 x^2+6 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{(-3+x)^2} \, dx-\frac {5}{3} \int \frac {x^2 \log (x)}{(-3+x)^2} \, dx+9 \int \frac {1}{(-3+x)^2 x} \, dx-\frac {28}{3} \int \frac {x^2}{(-3+x)^2} \, dx+10 \int \frac {x \log (x)}{(-3+x)^2} \, dx+52 \int \frac {x}{(-3+x)^2} \, dx\\ &=-\frac {15}{3-x}-\frac {1}{3} \int \left (\frac {e^{5-x} x \left (27-17 x+4 x^2\right )}{(-3+x)^2}+\frac {e^{5-x} x \left (6-4 x+x^2\right ) \log (x)}{(-3+x)^2}\right ) \, dx-\frac {5}{3} \int \left (\log (x)+\frac {9 \log (x)}{(-3+x)^2}+\frac {6 \log (x)}{-3+x}\right ) \, dx+9 \int \left (\frac {1}{3 (-3+x)^2}-\frac {1}{9 (-3+x)}+\frac {1}{9 x}\right ) \, dx-\frac {28}{3} \int \left (1+\frac {9}{(-3+x)^2}+\frac {6}{-3+x}\right ) \, dx+10 \int \left (\frac {3 \log (x)}{(-3+x)^2}+\frac {\log (x)}{-3+x}\right ) \, dx+52 \int \left (\frac {3}{(-3+x)^2}+\frac {1}{-3+x}\right ) \, dx\\ &=\frac {60}{3-x}-\frac {28 x}{3}-5 \log (3-x)+\log (x)-\frac {1}{3} \int \frac {e^{5-x} x \left (27-17 x+4 x^2\right )}{(-3+x)^2} \, dx-\frac {1}{3} \int \frac {e^{5-x} x \left (6-4 x+x^2\right ) \log (x)}{(-3+x)^2} \, dx-\frac {5}{3} \int \log (x) \, dx-15 \int \frac {\log (x)}{(-3+x)^2} \, dx+30 \int \frac {\log (x)}{(-3+x)^2} \, dx\\ &=\frac {60}{3-x}-\frac {23 x}{3}-5 \log (3-x)+\log (x)+e^{5-x} \log (x)-\frac {3 e^{5-x} \log (x)}{3-x}-\frac {5}{3} x \log (x)+\frac {1}{3} e^{5-x} x \log (x)+\frac {5 x \log (x)}{3-x}+\frac {1}{3} \int \frac {e^{5-x} x}{3-x} \, dx-\frac {1}{3} \int \left (7 e^{5-x}+\frac {36 e^{5-x}}{(-3+x)^2}+\frac {33 e^{5-x}}{-3+x}+4 e^{5-x} x\right ) \, dx-5 \int \frac {1}{-3+x} \, dx+10 \int \frac {1}{-3+x} \, dx\\ &=\frac {60}{3-x}-\frac {23 x}{3}+\log (x)+e^{5-x} \log (x)-\frac {3 e^{5-x} \log (x)}{3-x}-\frac {5}{3} x \log (x)+\frac {1}{3} e^{5-x} x \log (x)+\frac {5 x \log (x)}{3-x}+\frac {1}{3} \int \left (-e^{5-x}-\frac {3 e^{5-x}}{-3+x}\right ) \, dx-\frac {4}{3} \int e^{5-x} x \, dx-\frac {7}{3} \int e^{5-x} \, dx-11 \int \frac {e^{5-x}}{-3+x} \, dx-12 \int \frac {e^{5-x}}{(-3+x)^2} \, dx\\ &=\frac {7 e^{5-x}}{3}+\frac {60}{3-x}-\frac {12 e^{5-x}}{3-x}-\frac {23 x}{3}+\frac {4}{3} e^{5-x} x-11 e^2 \text {Ei}(3-x)+\log (x)+e^{5-x} \log (x)-\frac {3 e^{5-x} \log (x)}{3-x}-\frac {5}{3} x \log (x)+\frac {1}{3} e^{5-x} x \log (x)+\frac {5 x \log (x)}{3-x}-\frac {1}{3} \int e^{5-x} \, dx-\frac {4}{3} \int e^{5-x} \, dx+12 \int \frac {e^{5-x}}{-3+x} \, dx-\int \frac {e^{5-x}}{-3+x} \, dx\\ &=4 e^{5-x}+\frac {60}{3-x}-\frac {12 e^{5-x}}{3-x}-\frac {23 x}{3}+\frac {4}{3} e^{5-x} x+\log (x)+e^{5-x} \log (x)-\frac {3 e^{5-x} \log (x)}{3-x}-\frac {5}{3} x \log (x)+\frac {1}{3} e^{5-x} x \log (x)+\frac {5 x \log (x)}{3-x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.71, size = 62, normalized size = 2.00 \begin {gather*} \frac {e^{-x} \left (4 e^5 x^2+e^x \left (-180+69 x-23 x^2\right )+\left (e^5 x^2+e^x \left (-9+3 x-5 x^2\right )\right ) \log (x)\right )}{3 (-3+x)} \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. \(59\) vs.
\(2(28)=56\).
time = 0.42, size = 60, normalized size = 1.94
method | result | size |
norman | \(\frac {3 x -\frac {23 x^{2}}{3}-\frac {5 x^{2} \ln \left (x \right )}{3}+\frac {4 \,{\mathrm e}^{5-x} x^{2}}{3}+\frac {\ln \left (x \right ) {\mathrm e}^{5-x} x^{2}}{3}}{x -3}+\ln \left (x \right )\) | \(50\) |
default | \(\frac {\ln \left (x \right ) {\mathrm e}^{5-x} x^{2}+4 \,{\mathrm e}^{5-x} x^{2}}{3 x -9}-\frac {23 x}{3}-\frac {60}{x -3}+\ln \left (x \right )-\frac {5 x \ln \left (x \right )}{3}-\frac {5 x \ln \left (x \right )}{x -3}\) | \(60\) |
risch | \(\frac {\left ({\mathrm e}^{5-x} x^{2}-5 x^{2}+15 x -45\right ) \ln \left (x \right )}{3 x -9}-\frac {-4 \,{\mathrm e}^{5-x} x^{2}+12 x \ln \left (x \right )+23 x^{2}-36 \ln \left (x \right )-69 x +180}{3 \left (x -3\right )}\) | \(68\) |
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.
\(2 (28) = 56\).
time = 0.32, size = 64, normalized size = 2.06 \begin {gather*} -\frac {28}{3} \, x + \frac {5 \, x^{2} + {\left (x^{2} e^{5} \log \left (x\right ) + 4 \, x^{2} e^{5}\right )} e^{\left (-x\right )} - 5 \, {\left (x^{2} - 3 \, x + 9\right )} \log \left (x\right ) - 15 \, x}{3 \, {\left (x - 3\right )}} - \frac {60}{x - 3} - 4 \, \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 = 51, normalized size = 1.65 \begin {gather*} \frac {4 \, x^{2} e^{\left (-x + 5\right )} - 23 \, x^{2} + {\left (x^{2} e^{\left (-x + 5\right )} - 5 \, x^{2} + 3 \, x - 9\right )} \log \left (x\right ) + 69 \, x - 180}{3 \, {\left (x - 3\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. 54 vs.
\(2 (24) = 48\).
time = 0.18, size = 54, normalized size = 1.74 \begin {gather*} - \frac {23 x}{3} - 4 \log {\left (x \right )} + \frac {\left (x^{2} \log {\left (x \right )} + 4 x^{2}\right ) e^{5 - x}}{3 x - 9} + \frac {\left (- 5 x^{2} + 15 x - 45\right ) \log {\left (x \right )}}{3 x - 9} - \frac {60}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 56, normalized size = 1.81 \begin {gather*} \frac {x^{2} e^{\left (-x + 5\right )} \log \left (x\right ) + 4 \, x^{2} e^{\left (-x + 5\right )} - 5 \, x^{2} \log \left (x\right ) - 23 \, x^{2} + 3 \, x \log \left (x\right ) + 69 \, x - 9 \, \log \left (x\right ) - 180}{3 \, {\left (x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.10, size = 64, normalized size = 2.06 \begin {gather*} \ln \left (x\right )-\frac {23\,x}{3}-\frac {60}{x-3}+\frac {4\,x^2\,{\mathrm {e}}^{5-x}}{3\,\left (x-3\right )}-\frac {5\,x^2\,\ln \left (x\right )}{3\,\left (x-3\right )}+\frac {x^2\,{\mathrm {e}}^{5-x}\,\ln \left (x\right )}{3\,\left (x-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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