Optimal. Leaf size=61 \[ -6 x+\frac {6 (d+e x) \log (c (d+e x))}{e}-\frac {3 (d+e x) \log ^2(c (d+e x))}{e}+\frac {(d+e x) \log ^3(c (d+e x))}{e} \]
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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2436, 2333,
2332} \begin {gather*} \frac {(d+e x) \log ^3(c (d+e x))}{e}-\frac {3 (d+e x) \log ^2(c (d+e x))}{e}+\frac {6 (d+e x) \log (c (d+e x))}{e}-6 x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2436
Rubi steps
\begin {align*} \int \log ^3(c (d+e x)) \, dx &=\frac {\text {Subst}\left (\int \log ^3(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x) \log ^3(c (d+e x))}{e}-\frac {3 \text {Subst}\left (\int \log ^2(c x) \, dx,x,d+e x\right )}{e}\\ &=-\frac {3 (d+e x) \log ^2(c (d+e x))}{e}+\frac {(d+e x) \log ^3(c (d+e x))}{e}+\frac {6 \text {Subst}(\int \log (c x) \, dx,x,d+e x)}{e}\\ &=-6 x+\frac {6 (d+e x) \log (c (d+e x))}{e}-\frac {3 (d+e x) \log ^2(c (d+e x))}{e}+\frac {(d+e x) \log ^3(c (d+e x))}{e}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 57, normalized size = 0.93 \begin {gather*} \frac {-6 e x+6 (d+e x) \log (c (d+e x))-3 (d+e x) \log ^2(c (d+e x))+(d+e x) \log ^3(c (d+e x))}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 78, normalized size = 1.28
method | result | size |
risch | \(\frac {\left (e x +d \right ) \ln \left (c \left (e x +d \right )\right )^{3}}{e}-\frac {3 \left (e x +d \right ) \ln \left (c \left (e x +d \right )\right )^{2}}{e}+6 x \ln \left (c \left (e x +d \right )\right )-6 x +\frac {6 d \ln \left (e x +d \right )}{e}\) | \(67\) |
derivativedivides | \(\frac {\left (c e x +c d \right ) \ln \left (c e x +c d \right )^{3}-3 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{2}+6 \left (c e x +c d \right ) \ln \left (c e x +c d \right )-6 c e x -6 c d}{c e}\) | \(78\) |
default | \(\frac {\left (c e x +c d \right ) \ln \left (c e x +c d \right )^{3}-3 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{2}+6 \left (c e x +c d \right ) \ln \left (c e x +c d \right )-6 c e x -6 c d}{c e}\) | \(78\) |
norman | \(x \ln \left (c \left (e x +d \right )\right )^{3}+\frac {d \ln \left (c \left (e x +d \right )\right )^{3}}{e}-6 x +6 x \ln \left (c \left (e x +d \right )\right )-3 x \ln \left (c \left (e x +d \right )\right )^{2}+\frac {6 d \ln \left (c \left (e x +d \right )\right )}{e}-\frac {3 d \ln \left (c \left (e x +d \right )\right )^{2}}{e}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs.
\(2 (64) = 128\).
time = 0.28, size = 134, normalized size = 2.20 \begin {gather*} 3 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} e \log \left ({\left (x e + d\right )} c\right )^{2} + x \log \left ({\left (x e + d\right )} c\right )^{3} - {\left (3 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} e^{\left (-2\right )} \log \left ({\left (x e + d\right )} c\right ) - {\left (d \log \left (x e + d\right )^{3} + 3 \, d \log \left (x e + d\right )^{2} - 6 \, x e + 6 \, d \log \left (x e + d\right )\right )} e^{\left (-2\right )}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 66, normalized size = 1.08 \begin {gather*} {\left ({\left (x e + d\right )} \log \left (c x e + c d\right )^{3} - 3 \, {\left (x e + d\right )} \log \left (c x e + c d\right )^{2} - 6 \, x e + 6 \, {\left (x e + d\right )} \log \left (c x e + c d\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 68, normalized size = 1.11 \begin {gather*} - 6 e \left (- \frac {d \log {\left (d + e x \right )}}{e^{2}} + \frac {x}{e}\right ) + 6 x \log {\left (c \left (d + e x\right ) \right )} + \frac {\left (- 3 d - 3 e x\right ) \log {\left (c \left (d + e x\right ) \right )}^{2}}{e} + \frac {\left (d + e x\right ) \log {\left (c \left (d + e x\right ) \right )}^{3}}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.68, size = 71, normalized size = 1.16 \begin {gather*} {\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{3} - 3 \, {\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{2} + 6 \, {\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right ) - 6 \, {\left (x e + d\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 88, normalized size = 1.44 \begin {gather*} 6\,x\,\ln \left (c\,d+c\,e\,x\right )-6\,x-3\,x\,{\ln \left (c\,d+c\,e\,x\right )}^2+x\,{\ln \left (c\,d+c\,e\,x\right )}^3-\frac {3\,d\,{\ln \left (c\,d+c\,e\,x\right )}^2}{e}+\frac {d\,{\ln \left (c\,d+c\,e\,x\right )}^3}{e}+\frac {6\,d\,\ln \left (d+e\,x\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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