Integrand size = 10, antiderivative size = 41 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=\frac {9 \operatorname {ExpIntegralEi}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2343, 2346, 2209} \[ \int \frac {x^2}{\log ^3(c x)} \, dx=\frac {9 \operatorname {ExpIntegralEi}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]
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Rule 2209
Rule 2343
Rule 2346
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3}{2 \log ^2(c x)}+\frac {3}{2} \int \frac {x^2}{\log ^2(c x)} \, dx \\ & = -\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)}+\frac {9}{2} \int \frac {x^2}{\log (c x)} \, dx \\ & = -\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)}+\frac {9 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (c x)\right )}{2 c^3} \\ & = \frac {9 \text {Ei}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=\frac {9 \operatorname {ExpIntegralEi}(3 \log (c x))}{2 c^3}-\frac {x^3}{2 \log ^2(c x)}-\frac {3 x^3}{2 \log (c x)} \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {x^{3} \left (1+3 \ln \left (x c \right )\right )}{2 \ln \left (x c \right )^{2}}-\frac {9 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{2 c^{3}}\) | \(34\) |
derivativedivides | \(\frac {-\frac {x^{3} c^{3}}{2 \ln \left (x c \right )^{2}}-\frac {3 x^{3} c^{3}}{2 \ln \left (x c \right )}-\frac {9 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{2}}{c^{3}}\) | \(44\) |
default | \(\frac {-\frac {x^{3} c^{3}}{2 \ln \left (x c \right )^{2}}-\frac {3 x^{3} c^{3}}{2 \ln \left (x c \right )}-\frac {9 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{2}}{c^{3}}\) | \(44\) |
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Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=-\frac {3 \, c^{3} x^{3} \log \left (c x\right ) + c^{3} x^{3} - 9 \, \log \left (c x\right )^{2} \operatorname {log\_integral}\left (c^{3} x^{3}\right )}{2 \, c^{3} \log \left (c x\right )^{2}} \]
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\[ \int \frac {x^2}{\log ^3(c x)} \, dx=\frac {- 3 x^{3} \log {\left (c x \right )} - x^{3}}{2 \log {\left (c x \right )}^{2}} + \frac {9 \int \frac {x^{2}}{\log {\left (c x \right )}}\, dx}{2} \]
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Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.32 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=-\frac {9 \, \Gamma \left (-2, -3 \, \log \left (c x\right )\right )}{c^{3}} \]
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Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\log ^3(c x)} \, dx=-\frac {3 \, x^{3}}{2 \, \log \left (c x\right )} - \frac {x^{3}}{2 \, \log \left (c x\right )^{2}} + \frac {9 \, {\rm Ei}\left (3 \, \log \left (c x\right )\right )}{2 \, c^{3}} \]
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Timed out. \[ \int \frac {x^2}{\log ^3(c x)} \, dx=\int \frac {x^2}{{\ln \left (c\,x\right )}^3} \,d x \]
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