Optimal. Leaf size=44 \[ \frac {1}{3} \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )+\frac {1}{3} p \text {Li}_2\left (1+\frac {b x^3}{a}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2441,
2352} \begin {gather*} \frac {1}{3} p \text {PolyLog}\left (2,\frac {b x^3}{a}+1\right )+\frac {1}{3} \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )-\frac {1}{3} (b p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )+\frac {1}{3} p \text {Li}_2\left (1+\frac {b x^3}{a}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.98 \begin {gather*} \frac {1}{3} \left (\log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )+p \text {Li}_2\left (\frac {a+b x^3}{a}\right )\right ) \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 = 0.30, size = 180, normalized size = 4.09
method | result | size |
risch | \(\ln \left (x \right ) \ln \left (\left (x^{3} b +a \right )^{p}\right )-p \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )+\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2}}{2}-\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{2}-\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{3}}{2}+\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}+\ln \left (c \right ) \ln \left (x \right )\) | \(180\) |
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. 80 vs.
\(2 (39) = 78\).
time = 0.27, size = 80, normalized size = 1.82 \begin {gather*} \frac {1}{3} \, b p {\left (\frac {3 \, \log \left (b x^{3} + a\right ) \log \left (x\right )}{b} - \frac {3 \, \log \left (\frac {b x^{3}}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x^{3}}{a}\right )}{b}\right )} - p \log \left (b x^{3} + a\right ) \log \left (x\right ) + \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \log \left (x\right ) \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 (a + b x^{3}\right )^{p} \right )}}{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 = -0.02 \begin {gather*} \int \frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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