Integrand size = 10, antiderivative size = 39 \[ \int \frac {1}{x^2 \log ^3(c x)} \, dx=\frac {1}{2} c \operatorname {ExpIntegralEi}(-\log (c x))-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)} \]
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Time = 0.03 (sec) , antiderivative size = 39, 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} \[ \int \frac {1}{x^2 \log ^3(c x)} \, dx=\frac {1}{2} c \operatorname {ExpIntegralEi}(-\log (c x))-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)} \]
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Rule 2209
Rule 2343
Rule 2346
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 x \log ^2(c x)}-\frac {1}{2} \int \frac {1}{x^2 \log ^2(c x)} \, dx \\ & = -\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)}+\frac {1}{2} \int \frac {1}{x^2 \log (c x)} \, dx \\ & = -\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)}+\frac {1}{2} c \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (c x)\right ) \\ & = \frac {1}{2} c \text {Ei}(-\log (c x))-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \log ^3(c x)} \, dx=\frac {1}{2} c \operatorname {ExpIntegralEi}(-\log (c x))-\frac {1}{2 x \log ^2(c x)}+\frac {1}{2 x \log (c x)} \]
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Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {\ln \left (x c \right )-1}{2 x \ln \left (x c \right )^{2}}-\frac {c \,\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )}{2}\) | \(28\) |
derivativedivides | \(c \left (-\frac {1}{2 x c \ln \left (x c \right )^{2}}+\frac {1}{2 x c \ln \left (x c \right )}-\frac {\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )}{2}\right )\) | \(40\) |
default | \(c \left (-\frac {1}{2 x c \ln \left (x c \right )^{2}}+\frac {1}{2 x c \ln \left (x c \right )}-\frac {\operatorname {Ei}_{1}\left (\ln \left (x c \right )\right )}{2}\right )\) | \(40\) |
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Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^2 \log ^3(c x)} \, dx=\frac {c x \log \left (c x\right )^{2} \operatorname {log\_integral}\left (\frac {1}{c x}\right ) + \log \left (c x\right ) - 1}{2 \, x \log \left (c x\right )^{2}} \]
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\[ \int \frac {1}{x^2 \log ^3(c x)} \, dx=\frac {\int \frac {1}{x^{2} \log {\left (c x \right )}}\, dx}{2} + \frac {\log {\left (c x \right )} - 1}{2 x \log {\left (c x \right )}^{2}} \]
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Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \log ^3(c x)} \, dx=-c \Gamma \left (-2, \log \left (c x\right )\right ) \]
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\[ \int \frac {1}{x^2 \log ^3(c x)} \, dx=\int { \frac {1}{x^{2} \log \left (c x\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \log ^3(c x)} \, dx=\int \frac {1}{x^2\,{\ln \left (c\,x\right )}^3} \,d x \]
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