Optimal. Leaf size=113 \[ \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \text {Li}_3\left (1+\frac {e x^n}{d}\right )}{n}+\frac {6 p^3 \text {Li}_4\left (1+\frac {e x^n}{d}\right )}{n} \]
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Rubi [A]
time = 0.10, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2443,
2481, 2421, 2430, 6724} \begin {gather*} -\frac {6 p^2 \text {PolyLog}\left (3,\frac {e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {3 p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {6 p^3 \text {PolyLog}\left (4,\frac {e x^n}{d}+1\right )}{n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2430
Rule 2443
Rule 2481
Rule 2504
Rule 6724
Rubi steps
\begin {align*} \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {\log ^3\left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}-\frac {(3 e p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \log ^2\left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}-\frac {(3 p) \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {\left (6 p^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right ) \text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \text {Li}_3\left (1+\frac {e x^n}{d}\right )}{n}+\frac {\left (6 p^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \text {Li}_3\left (1+\frac {e x^n}{d}\right )}{n}+\frac {6 p^3 \text {Li}_4\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(113)=226\).
time = 0.11, size = 270, normalized size = 2.39 \begin {gather*} \frac {-n p^3 \log (x) \log ^3\left (d+e x^n\right )+p^3 \log \left (-\frac {e x^n}{d}\right ) \log ^3\left (d+e x^n\right )+3 n p^2 \log (x) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 p^2 \log \left (-\frac {e x^n}{d}\right ) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 n p \log (x) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+3 p \log \left (-\frac {e x^n}{d}\right ) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+n \log (x) \log ^3\left (c \left (d+e x^n\right )^p\right )+3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text {Li}_2\left (1+\frac {e x^n}{d}\right )-6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \text {Li}_3\left (1+\frac {e x^n}{d}\right )+6 p^3 \text {Li}_4\left (1+\frac {e x^n}{d}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 3.26, size = 6131, normalized size = 54.26
method | result | size |
risch | \(\text {Expression too large to display}\) | \(6131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{3}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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