Optimal. Leaf size=44 \[ \frac {1}{3} \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )+\frac {1}{3} p \text {Li}_2\left (\frac {b x^3}{a}+1\right ) \]
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Rubi [A] time = 0.05, 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 = {2454, 2394, 2315} \[ \frac {1}{3} p \text {PolyLog}\left (2,\frac {b x^3}{a}+1\right )+\frac {1}{3} \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right ) \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2394
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x} \, dx &=\frac {1}{3} \operatorname {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) \operatorname {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 \[ \frac {1}{3} \left (\log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )+p \text {Li}_2\left (\frac {b x^3+a}{a}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.41, size = 180, normalized size = 4.09 \[ -\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \ln \relax (x )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )^{2} \ln \relax (x )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )^{2} \ln \relax (x )}{2}-\frac {i \pi \mathrm {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )^{3} \ln \relax (x )}{2}-p \left (\ln \relax (x ) \ln \left (\frac {\RootOf \left (b \,\textit {\_Z}^{3}+a \right )-x}{\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}\right )+\dilog \left (\frac {\RootOf \left (b \,\textit {\_Z}^{3}+a \right )-x}{\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}\right )\right )+\ln \relax (c ) \ln \relax (x )+\ln \relax (x ) \ln \left (\left (b \,x^{3}+a \right )^{p}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 80, normalized size = 1.82 \[ \frac {1}{3} \, b p {\left (\frac {3 \, \log \left (b x^{3} + a\right ) \log \relax (x)}{b} - \frac {3 \, \log \left (\frac {b x^{3}}{a} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x^{3}}{a}\right )}{b}\right )} - p \log \left (b x^{3} + a\right ) \log \relax (x) + \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \log \relax (x) \]
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 \[ \int \frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x} \,d x \]
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 \[ \int \frac {\log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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