Integrand size = 8, antiderivative size = 37 \[ \int \frac {x}{\log ^3(c x)} \, dx=\frac {2 \operatorname {ExpIntegralEi}(2 \log (c x))}{c^2}-\frac {x^2}{2 \log ^2(c x)}-\frac {x^2}{\log (c x)} \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2343, 2346, 2209} \[ \int \frac {x}{\log ^3(c x)} \, dx=\frac {2 \operatorname {ExpIntegralEi}(2 \log (c x))}{c^2}-\frac {x^2}{2 \log ^2(c x)}-\frac {x^2}{\log (c x)} \]
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Rule 2209
Rule 2343
Rule 2346
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2}{2 \log ^2(c x)}+\int \frac {x}{\log ^2(c x)} \, dx \\ & = -\frac {x^2}{2 \log ^2(c x)}-\frac {x^2}{\log (c x)}+2 \int \frac {x}{\log (c x)} \, dx \\ & = -\frac {x^2}{2 \log ^2(c x)}-\frac {x^2}{\log (c x)}+\frac {2 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (c x)\right )}{c^2} \\ & = \frac {2 \text {Ei}(2 \log (c x))}{c^2}-\frac {x^2}{2 \log ^2(c x)}-\frac {x^2}{\log (c x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\log ^3(c x)} \, dx=\frac {2 \operatorname {ExpIntegralEi}(2 \log (c x))}{c^2}-\frac {x^2}{2 \log ^2(c x)}-\frac {x^2}{\log (c x)} \]
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Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92
method | result | size |
risch | \(-\frac {x^{2} \left (1+2 \ln \left (x c \right )\right )}{2 \ln \left (x c \right )^{2}}-\frac {2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x c \right )\right )}{c^{2}}\) | \(34\) |
derivativedivides | \(\frac {-\frac {x^{2} c^{2}}{2 \ln \left (x c \right )^{2}}-\frac {x^{2} c^{2}}{\ln \left (x c \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x c \right )\right )}{c^{2}}\) | \(44\) |
default | \(\frac {-\frac {x^{2} c^{2}}{2 \ln \left (x c \right )^{2}}-\frac {x^{2} c^{2}}{\ln \left (x c \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x c \right )\right )}{c^{2}}\) | \(44\) |
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Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int \frac {x}{\log ^3(c x)} \, dx=-\frac {2 \, c^{2} x^{2} \log \left (c x\right ) + c^{2} x^{2} - 4 \, \log \left (c x\right )^{2} \operatorname {log\_integral}\left (c^{2} x^{2}\right )}{2 \, c^{2} \log \left (c x\right )^{2}} \]
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\[ \int \frac {x}{\log ^3(c x)} \, dx=\frac {- 2 x^{2} \log {\left (c x \right )} - x^{2}}{2 \log {\left (c x \right )}^{2}} + 2 \int \frac {x}{\log {\left (c x \right )}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.35 \[ \int \frac {x}{\log ^3(c x)} \, dx=-\frac {4 \, \Gamma \left (-2, -2 \, \log \left (c x\right )\right )}{c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\log ^3(c x)} \, dx=-\frac {x^{2}}{\log \left (c x\right )} - \frac {x^{2}}{2 \, \log \left (c x\right )^{2}} + \frac {2 \, {\rm Ei}\left (2 \, \log \left (c x\right )\right )}{c^{2}} \]
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Timed out. \[ \int \frac {x}{\log ^3(c x)} \, dx=\int \frac {x}{{\ln \left (c\,x\right )}^3} \,d x \]
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