Integrand size = 6, antiderivative size = 18 \[ \int \frac {1}{\log ^2(c x)} \, dx=-\frac {x}{\log (c x)}+\frac {\operatorname {LogIntegral}(c x)}{c} \]
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Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2334, 2335} \[ \int \frac {1}{\log ^2(c x)} \, dx=\frac {\operatorname {LogIntegral}(c x)}{c}-\frac {x}{\log (c x)} \]
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Rule 2334
Rule 2335
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{\log (c x)}+\int \frac {1}{\log (c x)} \, dx \\ & = -\frac {x}{\log (c x)}+\frac {\text {li}(c x)}{c} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\log ^2(c x)} \, dx=-\frac {x}{\log (c x)}+\frac {\operatorname {LogIntegral}(c x)}{c} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33
method | result | size |
risch | \(-\frac {x}{\ln \left (x c \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{c}\) | \(24\) |
derivativedivides | \(\frac {-\frac {x c}{\ln \left (x c \right )}-\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{c}\) | \(26\) |
default | \(\frac {-\frac {x c}{\ln \left (x c \right )}-\operatorname {Ei}_{1}\left (-\ln \left (x c \right )\right )}{c}\) | \(26\) |
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none
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\log ^2(c x)} \, dx=-\frac {c x - \log \left (c x\right ) \operatorname {log\_integral}\left (c x\right )}{c \log \left (c x\right )} \]
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Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\log ^2(c x)} \, dx=- \frac {x}{\log {\left (c x \right )}} + \frac {\operatorname {li}{\left (c x \right )}}{c} \]
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none
Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\log ^2(c x)} \, dx=\frac {\Gamma \left (-1, -\log \left (c x\right )\right )}{c} \]
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none
Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\log ^2(c x)} \, dx=\frac {{\rm Ei}\left (\log \left (c x\right )\right )}{c} - \frac {x}{\log \left (c x\right )} \]
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\log ^2(c x)} \, dx=\frac {\mathrm {logint}\left (c\,x\right )}{c}-\frac {x}{\ln \left (c\,x\right )} \]
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