Optimal. Leaf size=24 \[ \frac{1}{10} \text{PolyLog}\left (2,a x^5\right )-\frac{1}{10} \text{PolyLog}\left (2,-a x^5\right ) \]
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Rubi [A] time = 0.0212865, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6095, 5912} \[ \frac{1}{10} \text{PolyLog}\left (2,a x^5\right )-\frac{1}{10} \text{PolyLog}\left (2,-a x^5\right ) \]
Antiderivative was successfully verified.
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Rule 6095
Rule 5912
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}\left (a x^5\right )}{x} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=-\frac{1}{10} \text{Li}_2\left (-a x^5\right )+\frac{\text{Li}_2\left (a x^5\right )}{10}\\ \end{align*}
Mathematica [A] time = 0.0112493, size = 22, normalized size = 0.92 \[ \frac{1}{10} \left (\text{PolyLog}\left (2,a x^5\right )-\text{PolyLog}\left (2,-a x^5\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.095, size = 85, normalized size = 3.5 \begin{align*} \ln \left ( x \right ){\it Artanh} \left ( a{x}^{5} \right ) +{\frac{1}{2}\sum _{{\it \_R1}={\it RootOf} \left ( a{{\it \_Z}}^{5}-1 \right ) }\ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) }-{\frac{1}{2}\sum _{{\it \_R1}={\it RootOf} \left ( a{{\it \_Z}}^{5}+1 \right ) }\ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.989419, size = 140, normalized size = 5.83 \begin{align*} -\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 \left (x\right ) - \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{artanh}\left (a x^{5}\right ) \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{artanh}\left (a x^{5}\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (a x^{5}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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