Optimal. Leaf size=85 \[ \frac{\text{li}(c (d+e x))}{6 c e}-\frac{d+e x}{6 e \log ^2(c (d+e x))}-\frac{d+e x}{3 e \log ^3(c (d+e x))}-\frac{d+e x}{6 e \log (c (d+e x))} \]
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Rubi [A] time = 0.032716, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2297, 2298} \[ \frac{\text{li}(c (d+e x))}{6 c e}-\frac{d+e x}{6 e \log ^2(c (d+e x))}-\frac{d+e x}{3 e \log ^3(c (d+e x))}-\frac{d+e x}{6 e \log (c (d+e x))} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2297
Rule 2298
Rubi steps
\begin{align*} \int \frac{1}{\log ^4(c (d+e x))} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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*}
Mathematica [A] time = 0.0186455, size = 57, normalized size = 0.67 \[ \frac{\frac{\text{li}(c (d+e x))}{c}-\frac{(d+e x) \left (\log ^2(c (d+e x))+\log (c (d+e x))+2\right )}{\log ^3(c (d+e x))}}{6 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 116, normalized size = 1.4 \begin{align*} -{\frac{x}{3\, \left ( \ln \left ( cex+cd \right ) \right ) ^{3}}}-{\frac{d}{3\,e \left ( \ln \left ( cex+cd \right ) \right ) ^{3}}}-{\frac{x}{6\, \left ( \ln \left ( cex+cd \right ) \right ) ^{2}}}-{\frac{d}{6\,e \left ( \ln \left ( cex+cd \right ) \right ) ^{2}}}-{\frac{x}{6\,\ln \left ( cex+cd \right ) }}-{\frac{d}{6\,e\ln \left ( cex+cd \right ) }}-{\frac{{\it Ei} \left ( 1,-\ln \left ( cex+cd \right ) \right ) }{6\,ce}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14629, size = 27, normalized size = 0.32 \begin{align*} \frac{\Gamma \left (-3, -\log \left (c e x + c d\right )\right )}{c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92004, size = 220, normalized size = 2.59 \begin{align*} \frac{\log \left (c e x + c d\right )^{3} \logintegral \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.860454, size = 71, normalized size = 0.84 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21198, size = 109, normalized size = 1.28 \begin{align*} \frac{{\rm Ei}\left (\log \left ({\left (x e + d\right )} c\right )\right ) e^{\left (-1\right )}}{6 \, c} - \frac{{\left (x e + d\right )} e^{\left (-1\right )}}{6 \, \log \left ({\left (x e + d\right )} c\right )} - \frac{{\left (x e + d\right )} e^{\left (-1\right )}}{6 \, \log \left ({\left (x e + d\right )} c\right )^{2}} - \frac{{\left (x e + d\right )} e^{\left (-1\right )}}{3 \, \log \left ({\left (x e + d\right )} c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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