Integrand size = 10, antiderivative size = 63 \[ \int \frac {1}{\log ^3(c (d+e x))} \, dx=-\frac {d+e x}{2 e \log ^2(c (d+e x))}-\frac {d+e x}{2 e \log (c (d+e x))}+\frac {\operatorname {LogIntegral}(c (d+e x))}{2 c e} \]
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Time = 0.02 (sec) , antiderivative size = 63, 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 = {2436, 2334, 2335} \[ \int \frac {1}{\log ^3(c (d+e x))} \, dx=\frac {\operatorname {LogIntegral}(c (d+e x))}{2 c e}-\frac {d+e x}{2 e \log ^2(c (d+e x))}-\frac {d+e x}{2 e \log (c (d+e x))} \]
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Rule 2334
Rule 2335
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\log ^3(c x)} \, dx,x,d+e x\right )}{e} \\ & = -\frac {d+e x}{2 e \log ^2(c (d+e x))}+\frac {\text {Subst}\left (\int \frac {1}{\log ^2(c x)} \, dx,x,d+e x\right )}{2 e} \\ & = -\frac {d+e x}{2 e \log ^2(c (d+e x))}-\frac {d+e x}{2 e \log (c (d+e x))}+\frac {\text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,d+e x\right )}{2 e} \\ & = -\frac {d+e x}{2 e \log ^2(c (d+e x))}-\frac {d+e x}{2 e \log (c (d+e x))}+\frac {\text {li}(c (d+e x))}{2 c e} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\log ^3(c (d+e x))} \, dx=\frac {-\frac {(d+e x) (1+\log (c (d+e x)))}{\log ^2(c (d+e x))}+\frac {\operatorname {LogIntegral}(c (d+e x))}{c}}{2 e} \]
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Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02
method | result | size |
risch | \(-\frac {\ln \left (c \left (e x +d \right )\right ) x e +d \ln \left (c \left (e x +d \right )\right )+e x +d}{2 e \ln \left (c \left (e x +d \right )\right )^{2}}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c e x +c d \right )\right )}{2 c e}\) | \(64\) |
derivativedivides | \(\frac {-\frac {c e x +c d}{2 \ln \left (c e x +c d \right )^{2}}-\frac {c e x +c d}{2 \ln \left (c e x +c d \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c e x +c d \right )\right )}{2}}{c e}\) | \(66\) |
default | \(\frac {-\frac {c e x +c d}{2 \ln \left (c e x +c d \right )^{2}}-\frac {c e x +c d}{2 \ln \left (c e x +c d \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c e x +c d \right )\right )}{2}}{c e}\) | \(66\) |
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Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\log ^3(c (d+e x))} \, dx=-\frac {c e x - \log \left (c e x + c d\right )^{2} \operatorname {log\_integral}\left (c e x + c d\right ) + c d + {\left (c e x + c d\right )} \log \left (c e x + c d\right )}{2 \, c e \log \left (c e x + c d\right )^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\log ^3(c (d+e x))} \, dx=\frac {- d - e x + \left (- d - e x\right ) \log {\left (c \left (d + e x\right ) \right )}}{2 e \log {\left (c \left (d + e x\right ) \right )}^{2}} + \frac {\operatorname {li}{\left (c d + c e x \right )}}{2 c e} \]
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\log ^3(c (d+e x))} \, dx=-\frac {\Gamma \left (-2, -\log \left (c e x + c d\right )\right )}{c e} \]
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Time = 0.34 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\log ^3(c (d+e x))} \, dx=\frac {{\rm Ei}\left (\log \left ({\left (e x + d\right )} c\right )\right )}{2 \, c e} - \frac {e x + d}{2 \, e \log \left ({\left (e x + d\right )} c\right )} - \frac {e x + d}{2 \, e \log \left ({\left (e x + d\right )} c\right )^{2}} \]
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Time = 1.50 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\log ^3(c (d+e x))} \, dx=\frac {\mathrm {logint}\left (c\,\left (d+e\,x\right )\right )}{2\,c\,e}-\frac {\frac {c\,d}{2}+\ln \left (c\,\left (d+e\,x\right )\right )\,\left (\frac {c\,d}{2}+\frac {c\,e\,x}{2}\right )+\frac {c\,e\,x}{2}}{c\,e\,{\ln \left (c\,\left (d+e\,x\right )\right )}^2} \]
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