Integrand size = 79, antiderivative size = 21 \[ \int \frac {-4 x^7+8 x^7 \log (4 x)+\left (-4 x^4-20 x^5\right ) \log ^2(4 x)+\left (10 x^4+60 x^5\right ) \log ^3(4 x)+\left (2 x+30 x^2+100 x^3\right ) \log ^5(4 x)}{\log ^5(4 x)} \, dx=x^2 \left (1+5 x+\frac {x^3}{\log ^2(4 x)}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).
Time = 0.42 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19, number of steps used = 33, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used = {6874, 14, 2343, 2346, 2209, 2395} \[ \int \frac {-4 x^7+8 x^7 \log (4 x)+\left (-4 x^4-20 x^5\right ) \log ^2(4 x)+\left (10 x^4+60 x^5\right ) \log ^3(4 x)+\left (2 x+30 x^2+100 x^3\right ) \log ^5(4 x)}{\log ^5(4 x)} \, dx=\frac {x^8}{\log ^4(4 x)}+\frac {10 x^6}{\log ^2(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+25 x^4+10 x^3+x^2 \]
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Rule 14
Rule 2209
Rule 2343
Rule 2346
Rule 2395
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (2 x \left (1+15 x+50 x^2\right )-\frac {4 x^7}{\log ^5(4 x)}+\frac {8 x^7}{\log ^4(4 x)}-\frac {4 x^4 (1+5 x)}{\log ^3(4 x)}+\frac {10 x^4 (1+6 x)}{\log ^2(4 x)}\right ) \, dx \\ & = 2 \int x \left (1+15 x+50 x^2\right ) \, dx-4 \int \frac {x^7}{\log ^5(4 x)} \, dx-4 \int \frac {x^4 (1+5 x)}{\log ^3(4 x)} \, dx+8 \int \frac {x^7}{\log ^4(4 x)} \, dx+10 \int \frac {x^4 (1+6 x)}{\log ^2(4 x)} \, dx \\ & = \frac {x^8}{\log ^4(4 x)}-\frac {8 x^8}{3 \log ^3(4 x)}+2 \int \left (x+15 x^2+50 x^3\right ) \, dx-4 \int \left (\frac {x^4}{\log ^3(4 x)}+\frac {5 x^5}{\log ^3(4 x)}\right ) \, dx-8 \int \frac {x^7}{\log ^4(4 x)} \, dx+10 \int \left (\frac {x^4}{\log ^2(4 x)}+\frac {6 x^5}{\log ^2(4 x)}\right ) \, dx+\frac {64}{3} \int \frac {x^7}{\log ^3(4 x)} \, dx \\ & = x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}-\frac {32 x^8}{3 \log ^2(4 x)}-4 \int \frac {x^4}{\log ^3(4 x)} \, dx+10 \int \frac {x^4}{\log ^2(4 x)} \, dx-20 \int \frac {x^5}{\log ^3(4 x)} \, dx-\frac {64}{3} \int \frac {x^7}{\log ^3(4 x)} \, dx+60 \int \frac {x^5}{\log ^2(4 x)} \, dx+\frac {256}{3} \int \frac {x^7}{\log ^2(4 x)} \, dx \\ & = x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)}-\frac {10 x^5}{\log (4 x)}-\frac {60 x^6}{\log (4 x)}-\frac {256 x^8}{3 \log (4 x)}-10 \int \frac {x^4}{\log ^2(4 x)} \, dx+50 \int \frac {x^4}{\log (4 x)} \, dx-60 \int \frac {x^5}{\log ^2(4 x)} \, dx-\frac {256}{3} \int \frac {x^7}{\log ^2(4 x)} \, dx+360 \int \frac {x^5}{\log (4 x)} \, dx+\frac {2048}{3} \int \frac {x^7}{\log (4 x)} \, dx \\ & = x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)}+\frac {1}{96} \text {Subst}\left (\int \frac {e^{8 x}}{x} \, dx,x,\log (4 x)\right )+\frac {25}{512} \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (4 x)\right )+\frac {45}{512} \text {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (4 x)\right )-50 \int \frac {x^4}{\log (4 x)} \, dx-360 \int \frac {x^5}{\log (4 x)} \, dx-\frac {2048}{3} \int \frac {x^7}{\log (4 x)} \, dx \\ & = x^2+10 x^3+25 x^4+\frac {25}{512} \text {Ei}(5 \log (4 x))+\frac {45}{512} \text {Ei}(6 \log (4 x))+\frac {1}{96} \text {Ei}(8 \log (4 x))+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)}-\frac {1}{96} \text {Subst}\left (\int \frac {e^{8 x}}{x} \, dx,x,\log (4 x)\right )-\frac {25}{512} \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (4 x)\right )-\frac {45}{512} \text {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (4 x)\right ) \\ & = x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).
Time = 0.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {-4 x^7+8 x^7 \log (4 x)+\left (-4 x^4-20 x^5\right ) \log ^2(4 x)+\left (10 x^4+60 x^5\right ) \log ^3(4 x)+\left (2 x+30 x^2+100 x^3\right ) \log ^5(4 x)}{\log ^5(4 x)} \, dx=x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(21)=42\).
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19
method | result | size |
risch | \(25 x^{4}+10 x^{3}+x^{2}+\frac {x^{5} \left (x^{3}+10 x \ln \left (4 x \right )^{2}+2 \ln \left (4 x \right )^{2}\right )}{\ln \left (4 x \right )^{4}}\) | \(46\) |
derivativedivides | \(25 x^{4}+10 x^{3}+\frac {10 x^{6}}{\ln \left (4 x \right )^{2}}+\frac {x^{8}}{\ln \left (4 x \right )^{4}}+x^{2}+\frac {2 x^{5}}{\ln \left (4 x \right )^{2}}\) | \(47\) |
default | \(25 x^{4}+10 x^{3}+\frac {10 x^{6}}{\ln \left (4 x \right )^{2}}+\frac {x^{8}}{\ln \left (4 x \right )^{4}}+x^{2}+\frac {2 x^{5}}{\ln \left (4 x \right )^{2}}\) | \(47\) |
parts | \(25 x^{4}+10 x^{3}+\frac {10 x^{6}}{\ln \left (4 x \right )^{2}}+\frac {x^{8}}{\ln \left (4 x \right )^{4}}+x^{2}+\frac {2 x^{5}}{\ln \left (4 x \right )^{2}}\) | \(47\) |
parallelrisch | \(-\frac {-x^{8}-10 \ln \left (4 x \right )^{2} x^{6}-25 x^{4} \ln \left (4 x \right )^{4}-2 x^{5} \ln \left (4 x \right )^{2}-10 \ln \left (4 x \right )^{4} x^{3}-x^{2} \ln \left (4 x \right )^{4}}{\ln \left (4 x \right )^{4}}\) | \(70\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {-4 x^7+8 x^7 \log (4 x)+\left (-4 x^4-20 x^5\right ) \log ^2(4 x)+\left (10 x^4+60 x^5\right ) \log ^3(4 x)+\left (2 x+30 x^2+100 x^3\right ) \log ^5(4 x)}{\log ^5(4 x)} \, dx=\frac {x^{8} + {\left (25 \, x^{4} + 10 \, x^{3} + x^{2}\right )} \log \left (4 \, x\right )^{4} + 2 \, {\left (5 \, x^{6} + x^{5}\right )} \log \left (4 \, x\right )^{2}}{\log \left (4 \, x\right )^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {-4 x^7+8 x^7 \log (4 x)+\left (-4 x^4-20 x^5\right ) \log ^2(4 x)+\left (10 x^4+60 x^5\right ) \log ^3(4 x)+\left (2 x+30 x^2+100 x^3\right ) \log ^5(4 x)}{\log ^5(4 x)} \, dx=25 x^{4} + 10 x^{3} + x^{2} + \frac {x^{8} + \left (10 x^{6} + 2 x^{5}\right ) \log {\left (4 x \right )}^{2}}{\log {\left (4 x \right )}^{4}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.52 \[ \int \frac {-4 x^7+8 x^7 \log (4 x)+\left (-4 x^4-20 x^5\right ) \log ^2(4 x)+\left (10 x^4+60 x^5\right ) \log ^3(4 x)+\left (2 x+30 x^2+100 x^3\right ) \log ^5(4 x)}{\log ^5(4 x)} \, dx=25 \, x^{4} + 10 \, x^{3} + x^{2} + \frac {25}{512} \, \Gamma \left (-1, -5 \, \log \left (4 \, x\right )\right ) + \frac {45}{512} \, \Gamma \left (-1, -6 \, \log \left (4 \, x\right )\right ) + \frac {25}{256} \, \Gamma \left (-2, -5 \, \log \left (4 \, x\right )\right ) + \frac {45}{256} \, \Gamma \left (-2, -6 \, \log \left (4 \, x\right )\right ) + \frac {1}{16} \, \Gamma \left (-3, -8 \, \log \left (4 \, x\right )\right ) + \frac {1}{4} \, \Gamma \left (-4, -8 \, \log \left (4 \, x\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {-4 x^7+8 x^7 \log (4 x)+\left (-4 x^4-20 x^5\right ) \log ^2(4 x)+\left (10 x^4+60 x^5\right ) \log ^3(4 x)+\left (2 x+30 x^2+100 x^3\right ) \log ^5(4 x)}{\log ^5(4 x)} \, dx=25 \, x^{4} + \frac {x^{8}}{\log \left (4 \, x\right )^{4}} + \frac {10 \, x^{6}}{\log \left (4 \, x\right )^{2}} + 10 \, x^{3} + \frac {2 \, x^{5}}{\log \left (4 \, x\right )^{2}} + x^{2} \]
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Time = 12.79 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {-4 x^7+8 x^7 \log (4 x)+\left (-4 x^4-20 x^5\right ) \log ^2(4 x)+\left (10 x^4+60 x^5\right ) \log ^3(4 x)+\left (2 x+30 x^2+100 x^3\right ) \log ^5(4 x)}{\log ^5(4 x)} \, dx=\frac {x^8+2\,x^5\,{\ln \left (4\,x\right )}^2\,\left (5\,x+1\right )}{{\ln \left (4\,x\right )}^4}+x^2\,{\left (5\,x+1\right )}^2 \]
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