Integrand size = 70, antiderivative size = 29 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2 x^2 \left (3+x \left (1+\frac {2 x}{\log (x)}+\frac {3 (2+x)}{\log (x)}\right )^2\right ) \]
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Time = 0.51 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93, number of steps used = 38, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6873, 12, 6820, 6874, 45, 2395, 2343, 2346, 2209, 2403} \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=\frac {50 x^5}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {20 x^4}{\log (x)}+2 x^3+\frac {72 x^3}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+6 x^2 \]
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Rule 12
Rule 45
Rule 2209
Rule 2343
Rule 2346
Rule 2395
Rule 2403
Rule 6820
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \left (-72 x-120 x^2-50 x^3+96 x \log (x)+230 x^2 \log (x)+125 x^3 \log (x)+36 x \log ^2(x)+40 x^2 \log ^2(x)+6 \log ^3(x)+3 x \log ^3(x)\right )}{\log ^3(x)} \, dx \\ & = 2 \int \frac {x \left (-72 x-120 x^2-50 x^3+96 x \log (x)+230 x^2 \log (x)+125 x^3 \log (x)+36 x \log ^2(x)+40 x^2 \log ^2(x)+6 \log ^3(x)+3 x \log ^3(x)\right )}{\log ^3(x)} \, dx \\ & = 2 \int \frac {x \left (-2 x (6+5 x)^2+x \left (96+230 x+125 x^2\right ) \log (x)+4 x (9+10 x) \log ^2(x)+3 (2+x) \log ^3(x)\right )}{\log ^3(x)} \, dx \\ & = 2 \int \left (3 x (2+x)-\frac {2 x^2 (6+5 x)^2}{\log ^3(x)}+\frac {x^2 (6+5 x) (16+25 x)}{\log ^2(x)}+\frac {4 x^2 (9+10 x)}{\log (x)}\right ) \, dx \\ & = 2 \int \frac {x^2 (6+5 x) (16+25 x)}{\log ^2(x)} \, dx-4 \int \frac {x^2 (6+5 x)^2}{\log ^3(x)} \, dx+6 \int x (2+x) \, dx+8 \int \frac {x^2 (9+10 x)}{\log (x)} \, dx \\ & = 2 \int \left (\frac {96 x^2}{\log ^2(x)}+\frac {230 x^3}{\log ^2(x)}+\frac {125 x^4}{\log ^2(x)}\right ) \, dx-4 \int \left (\frac {36 x^2}{\log ^3(x)}+\frac {60 x^3}{\log ^3(x)}+\frac {25 x^4}{\log ^3(x)}\right ) \, dx+6 \int \left (2 x+x^2\right ) \, dx+8 \int \left (\frac {9 x^2}{\log (x)}+\frac {10 x^3}{\log (x)}\right ) \, dx \\ & = 6 x^2+2 x^3+72 \int \frac {x^2}{\log (x)} \, dx+80 \int \frac {x^3}{\log (x)} \, dx-100 \int \frac {x^4}{\log ^3(x)} \, dx-144 \int \frac {x^2}{\log ^3(x)} \, dx+192 \int \frac {x^2}{\log ^2(x)} \, dx-240 \int \frac {x^3}{\log ^3(x)} \, dx+250 \int \frac {x^4}{\log ^2(x)} \, dx+460 \int \frac {x^3}{\log ^2(x)} \, dx \\ & = 6 x^2+2 x^3+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}-\frac {192 x^3}{\log (x)}-\frac {460 x^4}{\log (x)}-\frac {250 x^5}{\log (x)}+72 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+80 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-216 \int \frac {x^2}{\log ^2(x)} \, dx-250 \int \frac {x^4}{\log ^2(x)} \, dx-480 \int \frac {x^3}{\log ^2(x)} \, dx+576 \int \frac {x^2}{\log (x)} \, dx+1250 \int \frac {x^4}{\log (x)} \, dx+1840 \int \frac {x^3}{\log (x)} \, dx \\ & = 6 x^2+2 x^3+72 \text {Ei}(3 \log (x))+80 \text {Ei}(4 \log (x))+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+\frac {20 x^4}{\log (x)}+576 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-648 \int \frac {x^2}{\log (x)} \, dx-1250 \int \frac {x^4}{\log (x)} \, dx+1250 \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+1840 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-1920 \int \frac {x^3}{\log (x)} \, dx \\ & = 6 x^2+2 x^3+648 \text {Ei}(3 \log (x))+1920 \text {Ei}(4 \log (x))+1250 \text {Ei}(5 \log (x))+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+\frac {20 x^4}{\log (x)}-648 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-1250 \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )-1920 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right ) \\ & = 6 x^2+2 x^3+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+\frac {20 x^4}{\log (x)} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=6 x^2+2 x^3+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+\frac {20 x^4}{\log (x)} \]
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Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38
method | result | size |
risch | \(2 x^{3}+6 x^{2}+\frac {2 x^{3} \left (25 x^{2}+10 x \ln \left (x \right )+60 x +12 \ln \left (x \right )+36\right )}{\ln \left (x \right )^{2}}\) | \(40\) |
norman | \(\frac {72 x^{3}+120 x^{4}+50 x^{5}+6 x^{2} \ln \left (x \right )^{2}+24 x^{3} \ln \left (x \right )+2 x^{3} \ln \left (x \right )^{2}+20 x^{4} \ln \left (x \right )}{\ln \left (x \right )^{2}}\) | \(54\) |
parallelrisch | \(\frac {72 x^{3}+120 x^{4}+50 x^{5}+6 x^{2} \ln \left (x \right )^{2}+24 x^{3} \ln \left (x \right )+2 x^{3} \ln \left (x \right )^{2}+20 x^{4} \ln \left (x \right )}{\ln \left (x \right )^{2}}\) | \(54\) |
default | \(2 x^{3}+6 x^{2}+\frac {20 x^{4}}{\ln \left (x \right )}+\frac {50 x^{5}}{\ln \left (x \right )^{2}}+\frac {24 x^{3}}{\ln \left (x \right )}+\frac {120 x^{4}}{\ln \left (x \right )^{2}}+\frac {72 x^{3}}{\ln \left (x \right )^{2}}\) | \(57\) |
parts | \(2 x^{3}+6 x^{2}+\frac {20 x^{4}}{\ln \left (x \right )}+\frac {50 x^{5}}{\ln \left (x \right )^{2}}+\frac {24 x^{3}}{\ln \left (x \right )}+\frac {120 x^{4}}{\ln \left (x \right )^{2}}+\frac {72 x^{3}}{\ln \left (x \right )^{2}}\) | \(57\) |
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Time = 0.43 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=\frac {2 \, {\left (25 \, x^{5} + 60 \, x^{4} + 36 \, x^{3} + {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (5 \, x^{4} + 6 \, x^{3}\right )} \log \left (x\right )\right )}}{\log \left (x\right )^{2}} \]
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Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2 x^{3} + 6 x^{2} + \frac {50 x^{5} + 120 x^{4} + 72 x^{3} + \left (20 x^{4} + 24 x^{3}\right ) \log {\left (x \right )}}{\log {\left (x \right )}^{2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.52 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2 \, x^{3} + 6 \, x^{2} + 80 \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) + 72 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) + 576 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + 1840 \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) + 1250 \, \Gamma \left (-1, -5 \, \log \left (x\right )\right ) + 1296 \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) + 3840 \, \Gamma \left (-2, -4 \, \log \left (x\right )\right ) + 2500 \, \Gamma \left (-2, -5 \, \log \left (x\right )\right ) \]
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Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2 \, x^{3} + \frac {50 \, x^{5}}{\log \left (x\right )^{2}} + \frac {20 \, x^{4}}{\log \left (x\right )} + 6 \, x^{2} + \frac {120 \, x^{4}}{\log \left (x\right )^{2}} + \frac {24 \, x^{3}}{\log \left (x\right )} + \frac {72 \, x^{3}}{\log \left (x\right )^{2}} \]
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Time = 13.72 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-144 x^2-240 x^3-100 x^4+\left (192 x^2+460 x^3+250 x^4\right ) \log (x)+\left (72 x^2+80 x^3\right ) \log ^2(x)+\left (12 x+6 x^2\right ) \log ^3(x)}{\log ^3(x)} \, dx=2\,x^2\,\left (x+3\right )+\frac {2\,x^2\,\left (25\,x^3+60\,x^2+36\,x\right )+2\,x^2\,\ln \left (x\right )\,\left (10\,x^2+12\,x\right )}{{\ln \left (x\right )}^2} \]
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