Optimal. Leaf size=33 \[ (x+5 \log (4-x)) \left (-3+x^2+\frac {\left (-1+\frac {\log (x)}{3}\right )^2}{5 x}\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.48, antiderivative size = 249, normalized size of antiderivative = 7.55, number of steps
used = 143, number of rules used = 38, integrand size = 105, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.362, Rules used = {1607,
6874, 1634, 14, 2461, 6820, 78, 2442, 36, 31, 29, 2436, 2332, 45, 2393, 2353, 2352, 2341, 2417,
2458, 2384, 2338, 2423, 2439, 2438, 2422, 2354, 2421, 6724, 2395, 2333, 2418, 2339, 30, 2342,
2425, 2379, 2430} \begin {gather*} \frac {1}{6} \text {Li}_2\left (1-\frac {x}{4}\right )+\frac {\text {Li}_2\left (\frac {4}{x}\right )}{18}+\frac {2 \text {Li}_2\left (\frac {x}{4}\right )}{9}+\frac {\text {Li}_3\left (\frac {4}{x}\right )}{18}-\frac {\text {Li}_3\left (\frac {x}{4}\right )}{18}+\frac {1}{18} \text {Li}_2\left (\frac {4}{x}\right ) \log (x)+\frac {1}{18} \text {Li}_2\left (\frac {x}{4}\right ) \log (x)+x^3+5 x^2 \log (4-x)-3 x-\frac {\log ^3(x)}{108}-\frac {1}{36} \log \left (1-\frac {4}{x}\right ) \log ^2(x)+\frac {1}{36} \log \left (1-\frac {x}{4}\right ) \log ^2(x)-\frac {\log ^2(x)}{180}+\frac {\log (4-x) \log ^2(x)}{9 x}-15 \log (4-x)-\frac {1}{6} \log (4) \log (x-4)-\frac {1}{18} \log \left (1-\frac {4}{x}\right ) \log (x)+\frac {2}{9} \log (4-x) \log (x)-\frac {2}{9} \log (4) \log (x)-\frac {2 \log (x)}{15}+\frac {\log (4-x)}{x}-\frac {2 \log (4-x) \log (x)}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 30
Rule 31
Rule 36
Rule 45
Rule 78
Rule 1607
Rule 1634
Rule 2332
Rule 2333
Rule 2338
Rule 2339
Rule 2341
Rule 2342
Rule 2352
Rule 2353
Rule 2354
Rule 2379
Rule 2384
Rule 2393
Rule 2395
Rule 2417
Rule 2418
Rule 2421
Rule 2422
Rule 2423
Rule 2425
Rule 2430
Rule 2436
Rule 2438
Rule 2439
Rule 2442
Rule 2458
Rule 2461
Rule 6724
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{x^2 (-180+45 x)} \, dx\\ &=\int \left (\frac {23 x-47 x^2-45 x^3-105 x^4+45 x^5+100 \log (4-x)-25 x \log (4-x)-600 x^3 \log (4-x)+150 x^4 \log (4-x)}{15 (-4+x) x^2}+\frac {2 \left (-19 x+x^2-80 \log (4-x)+20 x \log (4-x)\right ) \log (x)}{45 (-4+x) x^2}-\frac {(-x-4 \log (4-x)+x \log (4-x)) \log ^2(x)}{9 (-4+x) x^2}\right ) \, dx\\ &=\frac {2}{45} \int \frac {\left (-19 x+x^2-80 \log (4-x)+20 x \log (4-x)\right ) \log (x)}{(-4+x) x^2} \, dx+\frac {1}{15} \int \frac {23 x-47 x^2-45 x^3-105 x^4+45 x^5+100 \log (4-x)-25 x \log (4-x)-600 x^3 \log (4-x)+150 x^4 \log (4-x)}{(-4+x) x^2} \, dx-\frac {1}{9} \int \frac {(-x-4 \log (4-x)+x \log (4-x)) \log ^2(x)}{(-4+x) x^2} \, dx\\ &=\frac {2}{45} \int \left (\frac {\left (-19 x+x^2-80 \log (4-x)+20 x \log (4-x)\right ) \log (x)}{16 (-4+x)}-\frac {\left (-19 x+x^2-80 \log (4-x)+20 x \log (4-x)\right ) \log (x)}{4 x^2}-\frac {\left (-19 x+x^2-80 \log (4-x)+20 x \log (4-x)\right ) \log (x)}{16 x}\right ) \, dx+\frac {1}{15} \int \left (\frac {23-47 x-45 x^2-105 x^3+45 x^4}{(-4+x) x}+\frac {25 \left (-1+6 x^3\right ) \log (4-x)}{x^2}\right ) \, dx-\frac {1}{9} \int \left (\frac {(-x-4 \log (4-x)+x \log (4-x)) \log ^2(x)}{16 (-4+x)}-\frac {(-x-4 \log (4-x)+x \log (4-x)) \log ^2(x)}{4 x^2}-\frac {(-x-4 \log (4-x)+x \log (4-x)) \log ^2(x)}{16 x}\right ) \, dx\\ &=\frac {1}{360} \int \frac {\left (-19 x+x^2-80 \log (4-x)+20 x \log (4-x)\right ) \log (x)}{-4+x} \, dx-\frac {1}{360} \int \frac {\left (-19 x+x^2-80 \log (4-x)+20 x \log (4-x)\right ) \log (x)}{x} \, dx-\frac {1}{144} \int \frac {(-x-4 \log (4-x)+x \log (4-x)) \log ^2(x)}{-4+x} \, dx+\frac {1}{144} \int \frac {(-x-4 \log (4-x)+x \log (4-x)) \log ^2(x)}{x} \, dx-\frac {1}{90} \int \frac {\left (-19 x+x^2-80 \log (4-x)+20 x \log (4-x)\right ) \log (x)}{x^2} \, dx+\frac {1}{36} \int \frac {(-x-4 \log (4-x)+x \log (4-x)) \log ^2(x)}{x^2} \, dx+\frac {1}{15} \int \frac {23-47 x-45 x^2-105 x^3+45 x^4}{(-4+x) x} \, dx+\frac {5}{3} \int \frac {\left (-1+6 x^3\right ) \log (4-x)}{x^2} \, dx\\ &=\frac {5 \log (4-x)}{3 x}+5 x^2 \log (4-x)+\frac {1}{360} \int \left (\frac {(-19+x) x}{-4+x}+20 \log (4-x)\right ) \log (x) \, dx-\frac {1}{360} \int \frac {((-19+x) x+20 (-4+x) \log (4-x)) \log (x)}{x} \, dx-\frac {1}{144} \int \left (-\frac {x}{-4+x}+\log (4-x)\right ) \log ^2(x) \, dx+\frac {1}{144} \int \frac {(-x+(-4+x) \log (4-x)) \log ^2(x)}{x} \, dx-\frac {1}{90} \int \frac {((-19+x) x+20 (-4+x) \log (4-x)) \log (x)}{x^2} \, dx+\frac {1}{36} \int \frac {(-x+(-4+x) \log (4-x)) \log ^2(x)}{x^2} \, dx+\frac {1}{15} \int \left (255+\frac {3915}{4 (-4+x)}-\frac {23}{4 x}+75 x+45 x^2\right ) \, dx+\frac {5}{3} \int \frac {1+3 x^3}{(4-x) x} \, dx\\ &=17 x+\frac {5 x^2}{2}+x^3+\frac {261}{4} \log (4-x)+\frac {5 \log (4-x)}{3 x}+5 x^2 \log (4-x)-\frac {23 \log (x)}{60}+\frac {1}{360} \int \left (-\frac {19 x \log (x)}{-4+x}+\frac {x^2 \log (x)}{-4+x}+20 \log (4-x) \log (x)\right ) \, dx-\frac {1}{360} \int \left (-19 \log (x)+x \log (x)+20 \log (4-x) \log (x)-\frac {80 \log (4-x) \log (x)}{x}\right ) \, dx-\frac {1}{144} \int \left (-\frac {x \log ^2(x)}{-4+x}+\log (4-x) \log ^2(x)\right ) \, dx+\frac {1}{144} \int \left (-\log ^2(x)+\log (4-x) \log ^2(x)-\frac {4 \log (4-x) \log ^2(x)}{x}\right ) \, dx-\frac {1}{90} \int \left (\log (x)-\frac {19 \log (x)}{x}-\frac {80 \log (4-x) \log (x)}{x^2}+\frac {20 \log (4-x) \log (x)}{x}\right ) \, dx+\frac {1}{36} \int \left (-\frac {\log ^2(x)}{x}-\frac {4 \log (4-x) \log ^2(x)}{x^2}+\frac {\log (4-x) \log ^2(x)}{x}\right ) \, dx+\frac {5}{3} \int \left (-12-\frac {193}{4 (-4+x)}+\frac {1}{4 x}-3 x\right ) \, dx\\ &=-3 x+x^3-\frac {91}{6} \log (4-x)+\frac {5 \log (4-x)}{3 x}+5 x^2 \log (4-x)+\frac {\log (x)}{30}-\frac {1}{360} \int x \log (x) \, dx+\frac {1}{360} \int \frac {x^2 \log (x)}{-4+x} \, dx-\frac {1}{144} \int \log ^2(x) \, dx+\frac {1}{144} \int \frac {x \log ^2(x)}{-4+x} \, dx-\frac {1}{90} \int \log (x) \, dx-\frac {1}{36} \int \frac {\log ^2(x)}{x} \, dx+\frac {19}{360} \int \log (x) \, dx-\frac {19}{360} \int \frac {x \log (x)}{-4+x} \, dx-\frac {1}{9} \int \frac {\log (4-x) \log ^2(x)}{x^2} \, dx+\frac {19}{90} \int \frac {\log (x)}{x} \, dx+\frac {8}{9} \int \frac {\log (4-x) \log (x)}{x^2} \, dx\\ &=-\frac {73 x}{24}+\frac {x^2}{1440}+x^3-\frac {91}{6} \log (4-x)+\frac {17 \log (4-x)}{9 x}+5 x^2 \log (4-x)+\frac {\log (x)}{30}+\frac {1}{24} x \log (x)-\frac {1}{720} x^2 \log (x)+\frac {2}{9} \log (4-x) \log (x)-\frac {2 \log (4-x) \log (x)}{3 x}-\frac {7 \log ^2(x)}{60}-\frac {1}{144} x \log ^2(x)+\frac {\log (4-x) \log ^2(x)}{9 x}+\frac {1}{360} \int \left (4 \log (x)+\frac {16 \log (x)}{-4+x}+x \log (x)\right ) \, dx+\frac {1}{144} \int \left (\log ^2(x)+\frac {4 \log ^2(x)}{-4+x}\right ) \, dx+\frac {1}{72} \int \log (x) \, dx-\frac {1}{36} \text {Subst}\left (\int x^2 \, dx,x,\log (x)\right )-\frac {19}{360} \int \left (\log (x)+\frac {4 \log (x)}{-4+x}\right ) \, dx-\frac {1}{9} \int \left (-\frac {2}{(4-x) x}-\frac {2 \log (x)}{(4-x) x}-\frac {\log ^2(x)}{(4-x) x}\right ) \, dx-\frac {8}{9} \int \left (-\frac {\log (4-x)}{x^2}+\frac {\log (4-x)}{4 x}-\frac {\log (x)}{4 x}\right ) \, dx\\ &=-\frac {55 x}{18}+\frac {x^2}{1440}+x^3-\frac {91}{6} \log (4-x)+\frac {17 \log (4-x)}{9 x}+5 x^2 \log (4-x)+\frac {\log (x)}{30}+\frac {1}{18} x \log (x)-\frac {1}{720} x^2 \log (x)+\frac {2}{9} \log (4-x) \log (x)-\frac {2 \log (4-x) \log (x)}{3 x}-\frac {7 \log ^2(x)}{60}-\frac {1}{144} x \log ^2(x)+\frac {\log (4-x) \log ^2(x)}{9 x}-\frac {\log ^3(x)}{108}+\frac {1}{360} \int x \log (x) \, dx+\frac {1}{144} \int \log ^2(x) \, dx+\frac {1}{90} \int \log (x) \, dx+\frac {1}{36} \int \frac {\log ^2(x)}{-4+x} \, dx+\frac {2}{45} \int \frac {\log (x)}{-4+x} \, dx-\frac {19}{360} \int \log (x) \, dx+\frac {1}{9} \int \frac {\log ^2(x)}{(4-x) x} \, dx-\frac {19}{90} \int \frac {\log (x)}{-4+x} \, dx+\frac {2}{9} \int \frac {1}{(4-x) x} \, dx-\frac {2}{9} \int \frac {\log (4-x)}{x} \, dx+\frac {2}{9} \int \frac {\log (x)}{x} \, dx+\frac {2}{9} \int \frac {\log (x)}{(4-x) x} \, dx+\frac {8}{9} \int \frac {\log (4-x)}{x^2} \, dx\\ &=-\frac {217 x}{72}+x^3-\frac {91}{6} \log (4-x)+\frac {\log (4-x)}{x}+5 x^2 \log (4-x)-\frac {1}{6} \log (4) \log (-4+x)+\frac {\log (x)}{30}+\frac {1}{72} x \log (x)-\frac {2}{9} \log (4) \log (x)+\frac {2}{9} \log (4-x) \log (x)-\frac {2 \log (4-x) \log (x)}{3 x}-\frac {\log ^2(x)}{180}+\frac {\log (4-x) \log ^2(x)}{9 x}+\frac {1}{36} \log \left (1-\frac {x}{4}\right ) \log ^2(x)-\frac {\log ^3(x)}{108}-\frac {1}{72} \int \log (x) \, dx+\frac {1}{36} \int \frac {\log ^2(x)}{4-x} \, dx+\frac {1}{36} \int \frac {\log ^2(x)}{x} \, dx+\frac {2}{45} \int \frac {\log \left (\frac {x}{4}\right )}{-4+x} \, dx+\frac {1}{18} \int \frac {1}{4-x} \, dx+\frac {1}{18} \int \frac {1}{x} \, dx+\frac {1}{18} \int \frac {\log (x)}{4-x} \, dx+\frac {1}{18} \int \frac {\log (x)}{x} \, dx-\frac {1}{18} \int \frac {\log \left (1-\frac {x}{4}\right ) \log (x)}{x} \, dx-\frac {19}{90} \int \frac {\log \left (\frac {x}{4}\right )}{-4+x} \, dx-\frac {2}{9} \int \frac {\log \left (1-\frac {x}{4}\right )}{x} \, dx-\frac {8}{9} \int \frac {1}{(4-x) x} \, dx\\ &=-3 x+x^3-\frac {137}{9} \log (4-x)+\frac {\log (4-x)}{x}+5 x^2 \log (4-x)-\frac {1}{18} \log (4) \log (4-x)-\frac {1}{6} \log (4) \log (-4+x)+\frac {4 \log (x)}{45}-\frac {2}{9} \log (4) \log (x)+\frac {2}{9} \log (4-x) \log (x)-\frac {2 \log (4-x) \log (x)}{3 x}+\frac {\log ^2(x)}{45}+\frac {\log (4-x) \log ^2(x)}{9 x}-\frac {\log ^3(x)}{108}+\frac {1}{6} \text {Li}_2\left (1-\frac {x}{4}\right )+\frac {2 \text {Li}_2\left (\frac {x}{4}\right )}{9}+\frac {1}{18} \log (x) \text {Li}_2\left (\frac {x}{4}\right )+\frac {1}{36} \text {Subst}\left (\int x^2 \, dx,x,\log (x)\right )+\frac {1}{18} \int \frac {\log \left (\frac {x}{4}\right )}{4-x} \, dx+\frac {1}{18} \int \frac {\log \left (1-\frac {x}{4}\right ) \log (x)}{x} \, dx-\frac {1}{18} \int \frac {\text {Li}_2\left (\frac {x}{4}\right )}{x} \, dx-\frac {2}{9} \int \frac {1}{4-x} \, dx-\frac {2}{9} \int \frac {1}{x} \, dx\\ &=-3 x+x^3-15 \log (4-x)+\frac {\log (4-x)}{x}+5 x^2 \log (4-x)-\frac {1}{18} \log (4) \log (4-x)-\frac {1}{6} \log (4) \log (-4+x)-\frac {2 \log (x)}{15}-\frac {2}{9} \log (4) \log (x)+\frac {2}{9} \log (4-x) \log (x)-\frac {2 \log (4-x) \log (x)}{3 x}+\frac {\log ^2(x)}{45}+\frac {\log (4-x) \log ^2(x)}{9 x}+\frac {2}{9} \text {Li}_2\left (1-\frac {x}{4}\right )+\frac {2 \text {Li}_2\left (\frac {x}{4}\right )}{9}-\frac {\text {Li}_3\left (\frac {x}{4}\right )}{18}+\frac {1}{18} \int \frac {\text {Li}_2\left (\frac {x}{4}\right )}{x} \, dx\\ &=-3 x+x^3-15 \log (4-x)+\frac {\log (4-x)}{x}+5 x^2 \log (4-x)-\frac {1}{18} \log (4) \log (4-x)-\frac {1}{6} \log (4) \log (-4+x)-\frac {2 \log (x)}{15}-\frac {2}{9} \log (4) \log (x)+\frac {2}{9} \log (4-x) \log (x)-\frac {2 \log (4-x) \log (x)}{3 x}+\frac {\log ^2(x)}{45}+\frac {\log (4-x) \log ^2(x)}{9 x}+\frac {2}{9} \text {Li}_2\left (1-\frac {x}{4}\right )+\frac {2 \text {Li}_2\left (\frac {x}{4}\right )}{9}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 50, normalized size = 1.52 \begin {gather*} \frac {1}{45} \left (45 x \left (-3+x^2\right )-6 \log (x)+\log ^2(x)+\frac {5 \log (4-x) \left (9-135 x+45 x^3-6 \log (x)+\log ^2(x)\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.23, size = 155, normalized size = 4.70
method | result | size |
risch | \(\frac {\left (45 x^{3}+\ln \left (x \right )^{2}-6 \ln \left (x \right )+9\right ) \ln \left (-x +4\right )}{9 x}+\frac {\ln \left (x \right )^{2}}{45}+x^{3}-3 x -15 \ln \left (x -4\right )-\frac {2 \ln \left (x \right )}{15}\) | \(50\) |
default | \(-\frac {\left (-\ln \left (x \right )^{2}-2 \ln \left (x \right )-2\right ) \ln \left (-x +4\right )}{9 x}-\frac {11 \ln \left (x \right )}{20}+\frac {785 \ln \left (x -4\right )}{12}+\frac {\ln \left (x \right )^{2}}{45}-\frac {2 \ln \left (x \right ) \ln \left (-\frac {x}{4}+1\right )}{45}-\frac {2 \polylog \left (2, \frac {x}{4}\right )}{45}+\frac {2 \left (-20-20 \ln \left (x \right )\right ) \ln \left (-x +4\right )}{45 x}+\frac {2 \left (\ln \left (x \right )-\ln \left (\frac {x}{4}\right )\right ) \ln \left (-\frac {x}{4}+1\right )}{45}-\frac {2 \dilog \left (\frac {x}{4}\right )}{45}+5 \left (-x +4\right )^{2} \ln \left (-x +4\right )-3 x +120-40 \left (-x +4\right ) \ln \left (-x +4\right )+\frac {5 \ln \left (-x \right )}{12}+\frac {5 \left (-x +4\right ) \ln \left (-x +4\right )}{12 x}+x^{3}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 53, normalized size = 1.61 \begin {gather*} \frac {45 \, x^{4} + x \log \left (x\right )^{2} - 135 \, x^{2} - 6 \, x \log \left (x\right ) + 5 \, {\left (45 \, x^{3} + \log \left (x\right )^{2} - 135 \, x - 6 \, \log \left (x\right ) + 9\right )} \log \left (-x + 4\right )}{45 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (29) = 58\).
time = 0.31, size = 63, normalized size = 1.91 \begin {gather*} \frac {45 \, x^{4} + {\left (x + 5 \, \log \left (-x + 4\right )\right )} \log \left (x\right )^{2} - 135 \, x^{2} - 6 \, {\left (x + 5 \, \log \left (-x + 4\right )\right )} \log \left (x\right ) + 45 \, {\left (5 \, x^{3} - 15 \, x + 1\right )} \log \left (-x + 4\right )}{45 \, x} \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. 51 vs.
\(2 (24) = 48\).
time = 1.64, size = 51, normalized size = 1.55 \begin {gather*} x^{3} - 3 x + \frac {\log {\left (x \right )}^{2}}{45} - \frac {2 \log {\left (x \right )}}{15} - 15 \log {\left (x - 4 \right )} + \frac {\left (45 x^{3} + \log {\left (x \right )}^{2} - 6 \log {\left (x \right )} + 9\right ) \log {\left (4 - x \right )}}{9 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 57, normalized size = 1.73 \begin {gather*} x^{3} + \frac {1}{45} \, \log \left (x\right )^{2} + \frac {1}{9} \, {\left (45 \, x^{2} + \frac {\log \left (x\right )^{2}}{x} - \frac {6 \, \log \left (x\right )}{x} + \frac {9}{x}\right )} \log \left (-x + 4\right ) - 3 \, x - 15 \, \log \left (x - 4\right ) - \frac {2}{15} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 72, normalized size = 2.18 \begin {gather*} \frac {{\ln \left (x\right )}^2}{45}-15\,\ln \left (x-4\right )-\frac {2\,\ln \left (x\right )}{15}-3\,x+x^3+\frac {\ln \left (4-x\right )}{x}+5\,x^2\,\ln \left (4-x\right )+\frac {\ln \left (4-x\right )\,{\ln \left (x\right )}^2}{9\,x}-\frac {2\,\ln \left (4-x\right )\,\ln \left (x\right )}{3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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