Optimal. Leaf size=30 \[ a \log (x)-\frac {1}{6} b \text {PolyLog}\left (2,-c x^3\right )+\frac {1}{6} b \text {PolyLog}\left (2,c x^3\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6035, 6031}
\begin {gather*} a \log (x)-\frac {1}{6} b \text {Li}_2\left (-c x^3\right )+\frac {1}{6} b \text {Li}_2\left (c x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6031
Rule 6035
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{x} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^3\right )\\ &=a \log (x)-\frac {1}{6} b \text {Li}_2\left (-c x^3\right )+\frac {1}{6} b \text {Li}_2\left (c x^3\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.93 \begin {gather*} a \log (x)+\frac {1}{6} b \left (-\text {PolyLog}\left (2,-c x^3\right )+\text {PolyLog}\left (2,c x^3\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 = 0.04, size = 92, normalized size = 3.07
method | result | size |
default | \(a \ln \left (x \right )+b \ln \left (x \right ) \arctanh \left (c \,x^{3}\right )-\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (c \,\textit {\_Z}^{3}+1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}+\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (c \,\textit {\_Z}^{3}-1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}\) | \(92\) |
risch | \(a \ln \left (x \right )-\frac {\ln \left (x \right ) \ln \left (-c \,x^{3}+1\right ) b}{2}+\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (c \,\textit {\_Z}^{3}-1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}+\frac {\ln \left (x \right ) \ln \left (c \,x^{3}+1\right ) b}{2}-\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (c \,\textit {\_Z}^{3}+1\right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}\) | \(109\) |
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 {a + b \operatorname {atanh}{\left (c x^{3} \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 = -0.03 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x^3\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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