Optimal. Leaf size=85 \[ -\frac {d+e x}{3 e \log ^3(c (d+e x))}-\frac {d+e x}{6 e \log ^2(c (d+e x))}-\frac {d+e x}{6 e \log (c (d+e x))}+\frac {\text {li}(c (d+e x))}{6 c e} \]
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Rubi [A]
time = 0.02, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2436, 2334,
2335} \begin {gather*} \frac {\text {li}(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))} \end {gather*}
Antiderivative was successfully verified.
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Rule 2334
Rule 2335
Rule 2436
Rubi steps
\begin {align*} \int \frac {1}{\log ^4(c (d+e x))} \, dx &=\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*}
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Mathematica [A]
time = 0.01, size = 57, normalized size = 0.67 \begin {gather*} \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 {\text {li}(c (d+e x))}{c}}{6 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 87, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {-\frac {c e x +c d}{3 \ln \left (c e x +c d \right )^{3}}-\frac {c e x +c d}{6 \ln \left (c e x +c d \right )^{2}}-\frac {c e x +c d}{6 \ln \left (c e x +c d \right )}-\frac {\expIntegral \left (1, -\ln \left (c e x +c d \right )\right )}{6}}{c e}\) | \(87\) |
default | \(\frac {-\frac {c e x +c d}{3 \ln \left (c e x +c d \right )^{3}}-\frac {c e x +c d}{6 \ln \left (c e x +c d \right )^{2}}-\frac {c e x +c d}{6 \ln \left (c e x +c d \right )}-\frac {\expIntegral \left (1, -\ln \left (c e x +c d \right )\right )}{6}}{c e}\) | \(87\) |
risch | \(-\frac {e x \ln \left (c \left (e x +d \right )\right )^{2}+d \ln \left (c \left (e x +d \right )\right )^{2}+e x \ln \left (c \left (e x +d \right )\right )+d \ln \left (c \left (e x +d \right )\right )+2 e x +2 d}{6 e \ln \left (c \left (e x +d \right )\right )^{3}}-\frac {\expIntegral \left (1, -\ln \left (c e x +c d \right )\right )}{6 c e}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 20, normalized size = 0.24 \begin {gather*} \frac {e^{\left (-1\right )} \Gamma \left (-3, -\log \left (c x e + c d\right )\right )}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 97, normalized size = 1.14 \begin {gather*} \frac {{\left (\log \left (c x e + c d\right )^{3} \operatorname {log\_integral}\left (c x e + c d\right ) - 2 \, c x e - {\left (c x e + c d\right )} \log \left (c x e + c d\right )^{2} - 2 \, c d - {\left (c x e + c d\right )} \log \left (c x e + c d\right )\right )} e^{\left (-1\right )}}{6 \, c \log \left (c x e + c d\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 71, normalized size = 0.84 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.28, size = 81, normalized size = 0.95 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 67, normalized size = 0.79 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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