Optimal. Leaf size=28 \[ \frac {1}{10} \text {Li}_2\left (-\frac {1}{a x^5}\right )-\frac {1}{10} \text {Li}_2\left (\frac {1}{a x^5}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6096, 5913} \[ \frac {1}{10} \text {PolyLog}\left (2,-\frac {1}{a x^5}\right )-\frac {1}{10} \text {PolyLog}\left (2,\frac {1}{a x^5}\right ) \]
Antiderivative was successfully verified.
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Rule 5913
Rule 6096
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}\left (a x^5\right )}{x} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {\coth ^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=\frac {1}{10} \text {Li}_2\left (-\frac {1}{a x^5}\right )-\frac {1}{10} \text {Li}_2\left (\frac {1}{a x^5}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ \frac {1}{10} \left (\text {Li}_2\left (-\frac {1}{a x^5}\right )-\text {Li}_2\left (\frac {1}{a x^5}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x^{5}\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (a x^{5}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 85, normalized size = 3.04 \[ \ln \relax (x ) \mathrm {arccoth}\left (a \,x^{5}\right )-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (a \,\textit {\_Z}^{5}+1\right )}{\sum }\left (\ln \relax (x ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (a \,\textit {\_Z}^{5}-1\right )}{\sum }\left (\ln \relax (x ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 104, normalized size = 3.71 \[ -\frac {1}{2} \, a {\left (\frac {\log \left (a x^{5} + 1\right )}{a} - \frac {\log \left (a x^{5} - 1\right )}{a}\right )} \log \relax (x) - \frac {1}{10} \, a {\left (\frac {\log \left (a x^{5} - 1\right ) \log \left (a x^{5}\right ) + {\rm Li}_2\left (-a x^{5} + 1\right )}{a} - \frac {\log \left (a x^{5} + 1\right ) \log \left (-a x^{5}\right ) + {\rm Li}_2\left (a x^{5} + 1\right )}{a}\right )} + \operatorname {arcoth}\left (a x^{5}\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {acoth}\left (a\,x^5\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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