Integrand size = 10, antiderivative size = 85 \[ \int \frac {1}{\log ^4(c (d+e x))} \, dx=-\frac {d+e x}{3 e \log ^3(c (d+e x))}-\frac {d+e x}{6 e \log ^2(c (d+e x))}-\frac {d+e x}{6 e \log (c (d+e x))}+\frac {\operatorname {LogIntegral}(c (d+e x))}{6 c e} \]
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Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2436, 2334, 2335} \[ \int \frac {1}{\log ^4(c (d+e x))} \, dx=\frac {\operatorname {LogIntegral}(c (d+e x))}{6 c e}-\frac {d+e x}{3 e \log ^3(c (d+e x))}-\frac {d+e x}{6 e \log ^2(c (d+e x))}-\frac {d+e x}{6 e \log (c (d+e x))} \]
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Rule 2334
Rule 2335
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\log ^4(c x)} \, dx,x,d+e x\right )}{e} \\ & = -\frac {d+e x}{3 e \log ^3(c (d+e x))}+\frac {\text {Subst}\left (\int \frac {1}{\log ^3(c x)} \, dx,x,d+e x\right )}{3 e} \\ & = -\frac {d+e x}{3 e \log ^3(c (d+e x))}-\frac {d+e x}{6 e \log ^2(c (d+e x))}+\frac {\text {Subst}\left (\int \frac {1}{\log ^2(c x)} \, dx,x,d+e x\right )}{6 e} \\ & = -\frac {d+e x}{3 e \log ^3(c (d+e x))}-\frac {d+e x}{6 e \log ^2(c (d+e x))}-\frac {d+e x}{6 e \log (c (d+e x))}+\frac {\text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,d+e x\right )}{6 e} \\ & = -\frac {d+e x}{3 e \log ^3(c (d+e x))}-\frac {d+e x}{6 e \log ^2(c (d+e x))}-\frac {d+e x}{6 e \log (c (d+e x))}+\frac {\text {li}(c (d+e x))}{6 c e} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\log ^4(c (d+e x))} \, dx=\frac {-\frac {(d+e x) \left (2+\log (c (d+e x))+\log ^2(c (d+e x))\right )}{\log ^3(c (d+e x))}+\frac {\operatorname {LogIntegral}(c (d+e x))}{c}}{6 e} \]
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Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {-\frac {c e x +c d}{3 \ln \left (c e x +c d \right )^{3}}-\frac {c e x +c d}{6 \ln \left (c e x +c d \right )^{2}}-\frac {c e x +c d}{6 \ln \left (c e x +c d \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c e x +c d \right )\right )}{6}}{c e}\) | \(87\) |
default | \(\frac {-\frac {c e x +c d}{3 \ln \left (c e x +c d \right )^{3}}-\frac {c e x +c d}{6 \ln \left (c e x +c d \right )^{2}}-\frac {c e x +c d}{6 \ln \left (c e x +c d \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c e x +c d \right )\right )}{6}}{c e}\) | \(87\) |
risch | \(-\frac {x \ln \left (c \left (e x +d \right )\right )^{2} e +\ln \left (c \left (e x +d \right )\right )^{2} d +\ln \left (c \left (e x +d \right )\right ) x e +d \ln \left (c \left (e x +d \right )\right )+2 e x +2 d}{6 e \ln \left (c \left (e x +d \right )\right )^{3}}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (c e x +c d \right )\right )}{6 c e}\) | \(92\) |
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\log ^4(c (d+e x))} \, dx=\frac {\log \left (c e x + c d\right )^{3} \operatorname {log\_integral}\left (c e x + c d\right ) - 2 \, c e x - {\left (c e x + c d\right )} \log \left (c e x + c d\right )^{2} - 2 \, c d - {\left (c e x + c d\right )} \log \left (c e x + c d\right )}{6 \, c e \log \left (c e x + c d\right )^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\log ^4(c (d+e x))} \, dx=\frac {- d - e x + \left (- \frac {d}{2} - \frac {e x}{2}\right ) \log {\left (c \left (d + e x\right ) \right )}^{2} + \left (- \frac {d}{2} - \frac {e x}{2}\right ) \log {\left (c \left (d + e x\right ) \right )}}{3 e \log {\left (c \left (d + e x\right ) \right )}^{3}} + \frac {\operatorname {li}{\left (c d + c e x \right )}}{6 c e} \]
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\log ^4(c (d+e x))} \, dx=\frac {\Gamma \left (-3, -\log \left (c e x + c d\right )\right )}{c e} \]
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Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\log ^4(c (d+e x))} \, dx=\frac {{\rm Ei}\left (\log \left ({\left (e x + d\right )} c\right )\right )}{6 \, c e} - \frac {e x + d}{6 \, e \log \left ({\left (e x + d\right )} c\right )} - \frac {e x + d}{6 \, e \log \left ({\left (e x + d\right )} c\right )^{2}} - \frac {e x + d}{3 \, e \log \left ({\left (e x + d\right )} c\right )^{3}} \]
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Time = 1.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\log ^4(c (d+e x))} \, dx=-\frac {\left (d+e\,x\right )\,\left (\frac {1}{6\,\ln \left (c\,\left (d+e\,x\right )\right )}+\frac {1}{6\,{\ln \left (c\,\left (d+e\,x\right )\right )}^2}+\frac {1}{3\,{\ln \left (c\,\left (d+e\,x\right )\right )}^3}\right )}{e}-\frac {\mathrm {expint}\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}{6\,c\,e} \]
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