Integrand size = 10, antiderivative size = 9 \[ \int \frac {1}{x^2 \log (c x)} \, dx=c \operatorname {ExpIntegralEi}(-\log (c x)) \]
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Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2346, 2209} \[ \int \frac {1}{x^2 \log (c x)} \, dx=c \operatorname {ExpIntegralEi}(-\log (c x)) \]
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Rule 2209
Rule 2346
Rubi steps \begin{align*} \text {integral}& = c \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (c x)\right ) \\ & = c \text {Ei}(-\log (c x)) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \log (c x)} \, dx=c \operatorname {ExpIntegralEi}(-\log (c x)) \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(-c \,\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )\) | \(10\) |
default | \(-c \,\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )\) | \(10\) |
risch | \(-c \,\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )\) | \(10\) |
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none
Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \log (c x)} \, dx=c \operatorname {log\_integral}\left (\frac {1}{c x}\right ) \]
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\[ \int \frac {1}{x^2 \log (c x)} \, dx=\int \frac {1}{x^{2} \log {\left (c x \right )}}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \log (c x)} \, dx=c {\rm Ei}\left (-\log \left (c x\right )\right ) \]
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\[ \int \frac {1}{x^2 \log (c x)} \, dx=\int { \frac {1}{x^{2} \log \left (c x\right )} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \log (c x)} \, dx=\int \frac {1}{x^2\,\ln \left (c\,x\right )} \,d x \]
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