Integrand size = 10, antiderivative size = 24 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\frac {3 \operatorname {ExpIntegralEi}(3 \log (c x))}{c^3}-\frac {x^3}{\log (c x)} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2343, 2346, 2209} \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\frac {3 \operatorname {ExpIntegralEi}(3 \log (c x))}{c^3}-\frac {x^3}{\log (c x)} \]
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Rule 2209
Rule 2343
Rule 2346
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3}{\log (c x)}+3 \int \frac {x^2}{\log (c x)} \, dx \\ & = -\frac {x^3}{\log (c x)}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (c x)\right )}{c^3} \\ & = \frac {3 \text {Ei}(3 \log (c x))}{c^3}-\frac {x^3}{\log (c x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\frac {3 \operatorname {ExpIntegralEi}(3 \log (c x))}{c^3}-\frac {x^3}{\log (c x)} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08
method | result | size |
risch | \(-\frac {x^{3}}{\ln \left (x c \right )}-\frac {3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\) | \(26\) |
derivativedivides | \(\frac {-\frac {x^{3} c^{3}}{\ln \left (x c \right )}-3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\) | \(30\) |
default | \(\frac {-\frac {x^{3} c^{3}}{\ln \left (x c \right )}-3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\) | \(30\) |
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none
Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=-\frac {c^{3} x^{3} - 3 \, \log \left (c x\right ) \operatorname {log\_integral}\left (c^{3} x^{3}\right )}{c^{3} \log \left (c x\right )} \]
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\[ \int \frac {x^2}{\log ^2(c x)} \, dx=- \frac {x^{3}}{\log {\left (c x \right )}} + 3 \int \frac {x^{2}}{\log {\left (c x \right )}}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\frac {3 \, \Gamma \left (-1, -3 \, \log \left (c x\right )\right )}{c^{3}} \]
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none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=-\frac {x^{3}}{\log \left (c x\right )} + \frac {3 \, {\rm Ei}\left (3 \, \log \left (c x\right )\right )}{c^{3}} \]
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Timed out. \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\int \frac {x^2}{{\ln \left (c\,x\right )}^2} \,d x \]
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