Integrand size = 8, antiderivative size = 43 \[ \int \frac {1}{x^3 \log ^4(x)} \, dx=-\frac {4}{3} \operatorname {ExpIntegralEi}(-2 \log (x))-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)} \]
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Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2343, 2346, 2209} \[ \int \frac {1}{x^3 \log ^4(x)} \, dx=-\frac {4}{3} \operatorname {ExpIntegralEi}(-2 \log (x))-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)} \]
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Rule 2209
Rule 2343
Rule 2346
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3 x^2 \log ^3(x)}-\frac {2}{3} \int \frac {1}{x^3 \log ^3(x)} \, dx \\ & = -\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}+\frac {2}{3} \int \frac {1}{x^3 \log ^2(x)} \, dx \\ & = -\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)}-\frac {4}{3} \int \frac {1}{x^3 \log (x)} \, dx \\ & = -\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)}-\frac {4}{3} \text {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {4}{3} \operatorname {ExpIntegralEi}(-2 \log (x))-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \log ^4(x)} \, dx=-\frac {4}{3} \operatorname {ExpIntegralEi}(-2 \log (x))-\frac {1}{3 x^2 \log ^3(x)}+\frac {1}{3 x^2 \log ^2(x)}-\frac {2}{3 x^2 \log (x)} \]
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Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72
method | result | size |
risch | \(-\frac {2 \ln \left (x \right )^{2}-\ln \left (x \right )+1}{3 x^{2} \ln \left (x \right )^{3}}+\frac {4 \,\operatorname {Ei}_{1}\left (2 \ln \left (x \right )\right )}{3}\) | \(31\) |
default | \(-\frac {1}{3 x^{2} \ln \left (x \right )^{3}}+\frac {1}{3 x^{2} \ln \left (x \right )^{2}}-\frac {2}{3 x^{2} \ln \left (x \right )}+\frac {4 \,\operatorname {Ei}_{1}\left (2 \ln \left (x \right )\right )}{3}\) | \(37\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^3 \log ^4(x)} \, dx=-\frac {4 \, x^{2} \log \left (x\right )^{3} \operatorname {log\_integral}\left (\frac {1}{x^{2}}\right ) + 2 \, \log \left (x\right )^{2} - \log \left (x\right ) + 1}{3 \, x^{2} \log \left (x\right )^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^3 \log ^4(x)} \, dx=- \frac {4 \operatorname {Ei}{\left (- 2 \log {\left (x \right )} \right )}}{3} + \frac {- 2 \log {\left (x \right )}^{2} + \log {\left (x \right )} - 1}{3 x^{2} \log {\left (x \right )}^{3}} \]
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Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.19 \[ \int \frac {1}{x^3 \log ^4(x)} \, dx=-8 \, \Gamma \left (-3, 2 \, \log \left (x\right )\right ) \]
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\[ \int \frac {1}{x^3 \log ^4(x)} \, dx=\int { \frac {1}{x^{3} \log \left (x\right )^{4}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^3 \log ^4(x)} \, dx=-\frac {4\,\mathrm {ei}\left (-2\,\ln \left (x\right )\right )}{3}-\frac {\frac {2\,{\ln \left (x\right )}^2}{3}-\frac {\ln \left (x\right )}{3}+\frac {1}{3}}{x^2\,{\ln \left (x\right )}^3} \]
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