Optimal. Leaf size=54 \[ -\frac{1}{10} \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}\left (a x^5\right )}\right )+\frac{1}{10} \text{csch}^{-1}\left (a x^5\right )^2-\frac{1}{5} \text{csch}^{-1}\left (a x^5\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (a x^5\right )}\right ) \]
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Rubi [A] time = 0.100677, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6282, 5659, 3716, 2190, 2279, 2391} \[ -\frac{1}{10} \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}\left (a x^5\right )}\right )+\frac{1}{10} \text{csch}^{-1}\left (a x^5\right )^2-\frac{1}{5} \text{csch}^{-1}\left (a x^5\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (a x^5\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 6282
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\text{csch}^{-1}\left (a x^5\right )}{x} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{\text{csch}^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=-\left (\frac{1}{5} \operatorname{Subst}\left (\int \frac{\sinh ^{-1}\left (\frac{x}{a}\right )}{x} \, dx,x,\frac{1}{x^5}\right )\right )\\ &=-\left (\frac{1}{5} \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\text{csch}^{-1}\left (a x^5\right )\right )\right )\\ &=\frac{1}{10} \text{csch}^{-1}\left (a x^5\right )^2+\frac{2}{5} \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}\left (a x^5\right )\right )\\ &=\frac{1}{10} \text{csch}^{-1}\left (a x^5\right )^2-\frac{1}{5} \text{csch}^{-1}\left (a x^5\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (a x^5\right )}\right )+\frac{1}{5} \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}\left (a x^5\right )\right )\\ &=\frac{1}{10} \text{csch}^{-1}\left (a x^5\right )^2-\frac{1}{5} \text{csch}^{-1}\left (a x^5\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (a x^5\right )}\right )+\frac{1}{10} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \text{csch}^{-1}\left (a x^5\right )}\right )\\ &=\frac{1}{10} \text{csch}^{-1}\left (a x^5\right )^2-\frac{1}{5} \text{csch}^{-1}\left (a x^5\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (a x^5\right )}\right )-\frac{1}{10} \text{Li}_2\left (e^{2 \text{csch}^{-1}\left (a x^5\right )}\right )\\ \end{align*}
Mathematica [A] time = 0.0401994, size = 49, normalized size = 0.91 \[ \frac{1}{10} \left (\text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}\left (a x^5\right )}\right )-\text{csch}^{-1}\left (a x^5\right ) \left (\text{csch}^{-1}\left (a x^5\right )+2 \log \left (1-e^{-2 \text{csch}^{-1}\left (a x^5\right )}\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.279, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccsch} \left (a{x}^{5}\right )}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} 5 \, a^{2} \int \frac{x^{9} \log \left (x\right )}{a^{2} x^{10} +{\left (a^{2} x^{10} + 1\right )}^{\frac{3}{2}} + 1}\,{d x} - \frac{1}{2} \, \log \left (a^{2} x^{10} + 1\right ) \log \left (x\right ) - \log \left (a\right ) \log \left (x\right ) - \frac{5}{2} \, \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\sqrt{a^{2} x^{10} + 1} + 1\right ) - \frac{1}{20} \,{\rm Li}_2\left (-a^{2} x^{10}\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{arcsch}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsch}{\left (a x^{5} \right )}}{x}\, dx \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{arcsch}\left (a x^{5}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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