Integrand size = 57, antiderivative size = 15 \[ \int \frac {-4+16 x-24 x^2+16 x^3-4 x^4+\left (24-80 x+96 x^2-48 x^3+8 x^4\right ) \log (-3+x)}{(-3+x) \log ^3(-3+x)} \, dx=\frac {2 (1-x)^4}{\log ^2(-3+x)} \]
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Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(15)=30\).
Time = 0.52 (sec) , antiderivative size = 145, normalized size of antiderivative = 9.67, number of steps used = 48, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {6820, 12, 6874, 2458, 2395, 2334, 2335, 2339, 30, 2343, 2346, 2209, 2447, 2446, 2436, 2437} \[ \int \frac {-4+16 x-24 x^2+16 x^3-4 x^4+\left (24-80 x+96 x^2-48 x^3+8 x^4\right ) \log (-3+x)}{(-3+x) \log ^3(-3+x)} \, dx=\frac {2 (3-x)^4}{\log ^2(x-3)}-\frac {16 (3-x)^3}{\log ^2(x-3)}+\frac {48 (3-x)^2}{\log ^2(x-3)}-\frac {64 (3-x)}{\log ^2(x-3)}+\frac {32}{\log ^2(x-3)}+\frac {8 (3-x)^4}{\log (x-3)}-\frac {48 (3-x)^3}{\log (x-3)}+\frac {96 (3-x)^2}{\log (x-3)}-\frac {8 (1-x)^3 (3-x)}{\log (x-3)}-\frac {64 (3-x)}{\log (x-3)} \]
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Rule 12
Rule 30
Rule 2209
Rule 2334
Rule 2335
Rule 2339
Rule 2343
Rule 2346
Rule 2395
Rule 2436
Rule 2437
Rule 2446
Rule 2447
Rule 2458
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {4 (1-x)^3 (1-x+2 (-3+x) \log (-3+x))}{(3-x) \log ^3(-3+x)} \, dx \\ & = 4 \int \frac {(1-x)^3 (1-x+2 (-3+x) \log (-3+x))}{(3-x) \log ^3(-3+x)} \, dx \\ & = 4 \int \left (-\frac {(-1+x)^4}{(-3+x) \log ^3(-3+x)}+\frac {2 (-1+x)^3}{\log ^2(-3+x)}\right ) \, dx \\ & = -\left (4 \int \frac {(-1+x)^4}{(-3+x) \log ^3(-3+x)} \, dx\right )+8 \int \frac {(-1+x)^3}{\log ^2(-3+x)} \, dx \\ & = -\frac {8 (1-x)^3 (3-x)}{\log (-3+x)}-4 \text {Subst}\left (\int \frac {(2+x)^4}{x \log ^3(x)} \, dx,x,-3+x\right )+32 \int \frac {(-1+x)^3}{\log (-3+x)} \, dx-48 \int \frac {(-1+x)^2}{\log (-3+x)} \, dx \\ & = -\frac {8 (1-x)^3 (3-x)}{\log (-3+x)}-4 \text {Subst}\left (\int \left (\frac {32}{\log ^3(x)}+\frac {16}{x \log ^3(x)}+\frac {24 x}{\log ^3(x)}+\frac {8 x^2}{\log ^3(x)}+\frac {x^3}{\log ^3(x)}\right ) \, dx,x,-3+x\right )+32 \int \left (\frac {8}{\log (-3+x)}+\frac {12 (-3+x)}{\log (-3+x)}+\frac {6 (-3+x)^2}{\log (-3+x)}+\frac {(-3+x)^3}{\log (-3+x)}\right ) \, dx-48 \int \left (\frac {4}{\log (-3+x)}+\frac {4 (-3+x)}{\log (-3+x)}+\frac {(-3+x)^2}{\log (-3+x)}\right ) \, dx \\ & = -\frac {8 (1-x)^3 (3-x)}{\log (-3+x)}-4 \text {Subst}\left (\int \frac {x^3}{\log ^3(x)} \, dx,x,-3+x\right )+32 \int \frac {(-3+x)^3}{\log (-3+x)} \, dx-32 \text {Subst}\left (\int \frac {x^2}{\log ^3(x)} \, dx,x,-3+x\right )-48 \int \frac {(-3+x)^2}{\log (-3+x)} \, dx-64 \text {Subst}\left (\int \frac {1}{x \log ^3(x)} \, dx,x,-3+x\right )-96 \text {Subst}\left (\int \frac {x}{\log ^3(x)} \, dx,x,-3+x\right )-128 \text {Subst}\left (\int \frac {1}{\log ^3(x)} \, dx,x,-3+x\right )-192 \int \frac {1}{\log (-3+x)} \, dx-192 \int \frac {-3+x}{\log (-3+x)} \, dx+192 \int \frac {(-3+x)^2}{\log (-3+x)} \, dx+256 \int \frac {1}{\log (-3+x)} \, dx+384 \int \frac {-3+x}{\log (-3+x)} \, dx \\ & = -\frac {64 (3-x)}{\log ^2(-3+x)}+\frac {48 (3-x)^2}{\log ^2(-3+x)}-\frac {16 (3-x)^3}{\log ^2(-3+x)}+\frac {2 (3-x)^4}{\log ^2(-3+x)}-\frac {8 (1-x)^3 (3-x)}{\log (-3+x)}-8 \text {Subst}\left (\int \frac {x^3}{\log ^2(x)} \, dx,x,-3+x\right )+32 \text {Subst}\left (\int \frac {x^3}{\log (x)} \, dx,x,-3+x\right )-48 \text {Subst}\left (\int \frac {x^2}{\log ^2(x)} \, dx,x,-3+x\right )-48 \text {Subst}\left (\int \frac {x^2}{\log (x)} \, dx,x,-3+x\right )-64 \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (-3+x)\right )-64 \text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,-3+x\right )-96 \text {Subst}\left (\int \frac {x}{\log ^2(x)} \, dx,x,-3+x\right )-192 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-3+x\right )-192 \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,-3+x\right )+192 \text {Subst}\left (\int \frac {x^2}{\log (x)} \, dx,x,-3+x\right )+256 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-3+x\right )+384 \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,-3+x\right ) \\ & = \frac {32}{\log ^2(-3+x)}-\frac {64 (3-x)}{\log ^2(-3+x)}+\frac {48 (3-x)^2}{\log ^2(-3+x)}-\frac {16 (3-x)^3}{\log ^2(-3+x)}+\frac {2 (3-x)^4}{\log ^2(-3+x)}-\frac {64 (3-x)}{\log (-3+x)}-\frac {8 (1-x)^3 (3-x)}{\log (-3+x)}+\frac {96 (3-x)^2}{\log (-3+x)}-\frac {48 (3-x)^3}{\log (-3+x)}+\frac {8 (3-x)^4}{\log (-3+x)}+64 \text {li}(-3+x)+32 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (-3+x)\right )-32 \text {Subst}\left (\int \frac {x^3}{\log (x)} \, dx,x,-3+x\right )-48 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (-3+x)\right )-64 \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-3+x\right )-144 \text {Subst}\left (\int \frac {x^2}{\log (x)} \, dx,x,-3+x\right )-192 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (-3+x)\right )+192 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (-3+x)\right )-192 \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,-3+x\right )+384 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (-3+x)\right ) \\ & = 192 \text {Ei}(2 \log (-3+x))+144 \text {Ei}(3 \log (-3+x))+32 \text {Ei}(4 \log (-3+x))+\frac {32}{\log ^2(-3+x)}-\frac {64 (3-x)}{\log ^2(-3+x)}+\frac {48 (3-x)^2}{\log ^2(-3+x)}-\frac {16 (3-x)^3}{\log ^2(-3+x)}+\frac {2 (3-x)^4}{\log ^2(-3+x)}-\frac {64 (3-x)}{\log (-3+x)}-\frac {8 (1-x)^3 (3-x)}{\log (-3+x)}+\frac {96 (3-x)^2}{\log (-3+x)}-\frac {48 (3-x)^3}{\log (-3+x)}+\frac {8 (3-x)^4}{\log (-3+x)}-32 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (-3+x)\right )-144 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (-3+x)\right )-192 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (-3+x)\right ) \\ & = \frac {32}{\log ^2(-3+x)}-\frac {64 (3-x)}{\log ^2(-3+x)}+\frac {48 (3-x)^2}{\log ^2(-3+x)}-\frac {16 (3-x)^3}{\log ^2(-3+x)}+\frac {2 (3-x)^4}{\log ^2(-3+x)}-\frac {64 (3-x)}{\log (-3+x)}-\frac {8 (1-x)^3 (3-x)}{\log (-3+x)}+\frac {96 (3-x)^2}{\log (-3+x)}-\frac {48 (3-x)^3}{\log (-3+x)}+\frac {8 (3-x)^4}{\log (-3+x)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {-4+16 x-24 x^2+16 x^3-4 x^4+\left (24-80 x+96 x^2-48 x^3+8 x^4\right ) \log (-3+x)}{(-3+x) \log ^3(-3+x)} \, dx=\frac {2 (-1+x)^4}{\log ^2(-3+x)} \]
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Time = 0.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80
method | result | size |
risch | \(\frac {2 x^{4}-8 x^{3}+12 x^{2}-8 x +2}{\ln \left (-3+x \right )^{2}}\) | \(27\) |
norman | \(\frac {2 x^{4}-8 x^{3}+12 x^{2}-8 x +2}{\ln \left (-3+x \right )^{2}}\) | \(28\) |
parallelrisch | \(\frac {2 x^{4}-8 x^{3}+12 x^{2}-8 x +2}{\ln \left (-3+x \right )^{2}}\) | \(28\) |
derivativedivides | \(\frac {2 \left (-3+x \right )^{4}}{\ln \left (-3+x \right )^{2}}+\frac {16 \left (-3+x \right )^{3}}{\ln \left (-3+x \right )^{2}}+\frac {48 \left (-3+x \right )^{2}}{\ln \left (-3+x \right )^{2}}+\frac {-192+64 x}{\ln \left (-3+x \right )^{2}}+\frac {32}{\ln \left (-3+x \right )^{2}}\) | \(60\) |
default | \(\frac {2 \left (-3+x \right )^{4}}{\ln \left (-3+x \right )^{2}}+\frac {16 \left (-3+x \right )^{3}}{\ln \left (-3+x \right )^{2}}+\frac {48 \left (-3+x \right )^{2}}{\ln \left (-3+x \right )^{2}}+\frac {-192+64 x}{\ln \left (-3+x \right )^{2}}+\frac {32}{\ln \left (-3+x \right )^{2}}\) | \(60\) |
parts | \(\frac {2 \left (-3+x \right )^{4}}{\ln \left (-3+x \right )^{2}}+\frac {16 \left (-3+x \right )^{3}}{\ln \left (-3+x \right )^{2}}+\frac {48 \left (-3+x \right )^{2}}{\ln \left (-3+x \right )^{2}}+\frac {-192+64 x}{\ln \left (-3+x \right )^{2}}+\frac {32}{\ln \left (-3+x \right )^{2}}\) | \(60\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {-4+16 x-24 x^2+16 x^3-4 x^4+\left (24-80 x+96 x^2-48 x^3+8 x^4\right ) \log (-3+x)}{(-3+x) \log ^3(-3+x)} \, dx=\frac {2 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}}{\log \left (x - 3\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {-4+16 x-24 x^2+16 x^3-4 x^4+\left (24-80 x+96 x^2-48 x^3+8 x^4\right ) \log (-3+x)}{(-3+x) \log ^3(-3+x)} \, dx=\frac {2 x^{4} - 8 x^{3} + 12 x^{2} - 8 x + 2}{\log {\left (x - 3 \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (13) = 26\).
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.20 \[ \int \frac {-4+16 x-24 x^2+16 x^3-4 x^4+\left (24-80 x+96 x^2-48 x^3+8 x^4\right ) \log (-3+x)}{(-3+x) \log ^3(-3+x)} \, dx=\frac {2 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 12 \, \log \left (x - 3\right )\right )}}{\log \left (x - 3\right )^{2}} - \frac {24}{\log \left (x - 3\right )} + \frac {2}{\log \left (x - 3\right )^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {-4+16 x-24 x^2+16 x^3-4 x^4+\left (24-80 x+96 x^2-48 x^3+8 x^4\right ) \log (-3+x)}{(-3+x) \log ^3(-3+x)} \, dx=\frac {2 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}}{\log \left (x - 3\right )^{2}} \]
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Time = 13.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {-4+16 x-24 x^2+16 x^3-4 x^4+\left (24-80 x+96 x^2-48 x^3+8 x^4\right ) \log (-3+x)}{(-3+x) \log ^3(-3+x)} \, dx=\frac {2\,{\left (x-1\right )}^4}{{\ln \left (x-3\right )}^2} \]
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