Integrand size = 27, antiderivative size = 19 \[ \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{4 \log ^2(x)} \, dx=x+3 \left (18 x+\frac {x^4}{4 \log (x)}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {12, 6874, 2343, 2346, 2209} \[ \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{4 \log ^2(x)} \, dx=\frac {3 x^4}{4 \log (x)}+55 x \]
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Rule 12
Rule 2209
Rule 2343
Rule 2346
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{\log ^2(x)} \, dx \\ & = \frac {1}{4} \int \left (220-\frac {3 x^3}{\log ^2(x)}+\frac {12 x^3}{\log (x)}\right ) \, dx \\ & = 55 x-\frac {3}{4} \int \frac {x^3}{\log ^2(x)} \, dx+3 \int \frac {x^3}{\log (x)} \, dx \\ & = 55 x+\frac {3 x^4}{4 \log (x)}-3 \int \frac {x^3}{\log (x)} \, dx+3 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right ) \\ & = 55 x+3 \operatorname {ExpIntegralEi}(4 \log (x))+\frac {3 x^4}{4 \log (x)}-3 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right ) \\ & = 55 x+\frac {3 x^4}{4 \log (x)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{4 \log ^2(x)} \, dx=55 x+\frac {3 x^4}{4 \log (x)} \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
default | \(55 x +\frac {3 x^{4}}{4 \ln \left (x \right )}\) | \(14\) |
risch | \(55 x +\frac {3 x^{4}}{4 \ln \left (x \right )}\) | \(14\) |
parts | \(55 x +\frac {3 x^{4}}{4 \ln \left (x \right )}\) | \(14\) |
norman | \(\frac {\frac {3 x^{4}}{4}+55 x \ln \left (x \right )}{\ln \left (x \right )}\) | \(17\) |
parallelrisch | \(\frac {3 x^{4}+220 x \ln \left (x \right )}{4 \ln \left (x \right )}\) | \(18\) |
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none
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{4 \log ^2(x)} \, dx=\frac {3 \, x^{4} + 220 \, x \log \left (x\right )}{4 \, \log \left (x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{4 \log ^2(x)} \, dx=\frac {3 x^{4}}{4 \log {\left (x \right )}} + 55 x \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{4 \log ^2(x)} \, dx=55 \, x + 3 \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) - 3 \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{4 \log ^2(x)} \, dx=\frac {3 \, x^{4}}{4 \, \log \left (x\right )} + 55 \, x \]
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Time = 8.17 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {-3 x^3+12 x^3 \log (x)+220 \log ^2(x)}{4 \log ^2(x)} \, dx=55\,x+\frac {3\,x^4}{4\,\ln \left (x\right )} \]
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