Integrand size = 10, antiderivative size = 22 \[ \int \frac {1}{x^2 \log ^2(c x)} \, dx=-c \operatorname {ExpIntegralEi}(-\log (c x))-\frac {1}{x \log (c x)} \]
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Time = 0.03 (sec) , antiderivative size = 22, 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} \[ \int \frac {1}{x^2 \log ^2(c x)} \, dx=-c \operatorname {ExpIntegralEi}(-\log (c x))-\frac {1}{x \log (c x)} \]
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Rule 2209
Rule 2343
Rule 2346
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{x \log (c x)}-\int \frac {1}{x^2 \log (c x)} \, dx \\ & = -\frac {1}{x \log (c x)}-c \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (c x)\right ) \\ & = -c \text {Ei}(-\log (c x))-\frac {1}{x \log (c x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \log ^2(c x)} \, dx=-c \operatorname {ExpIntegralEi}(-\log (c x))-\frac {1}{x \log (c x)} \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {1}{x \ln \left (x c \right )}+c \,\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )\) | \(21\) |
derivativedivides | \(c \left (-\frac {1}{x c \ln \left (x c \right )}+\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )\right )\) | \(24\) |
default | \(c \left (-\frac {1}{x c \ln \left (x c \right )}+\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )\right )\) | \(24\) |
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Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^2 \log ^2(c x)} \, dx=-\frac {c x \log \left (c x\right ) \operatorname {log\_integral}\left (\frac {1}{c x}\right ) + 1}{x \log \left (c x\right )} \]
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\[ \int \frac {1}{x^2 \log ^2(c x)} \, dx=- \int \frac {1}{x^{2} \log {\left (c x \right )}}\, dx - \frac {1}{x \log {\left (c x \right )}} \]
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Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x^2 \log ^2(c x)} \, dx=-c \Gamma \left (-1, \log \left (c x\right )\right ) \]
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\[ \int \frac {1}{x^2 \log ^2(c x)} \, dx=\int { \frac {1}{x^{2} \log \left (c x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \log ^2(c x)} \, dx=\int \frac {1}{x^2\,{\ln \left (c\,x\right )}^2} \,d x \]
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