Optimal. Leaf size=54 \[ -\frac {1}{10} \text {Li}_2\left (e^{2 \text {csch}^{-1}\left (a x^5\right )}\right )+\frac {1}{10} \text {csch}^{-1}\left (a x^5\right )^2-\frac {1}{5} \text {csch}^{-1}\left (a x^5\right ) \log \left (1-e^{2 \text {csch}^{-1}\left (a x^5\right )}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 6282
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*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 0.91 \[ \frac {1}{10} \left (\text {Li}_2\left (e^{-2 \text {csch}^{-1}\left (a x^5\right )}\right )-\text {csch}^{-1}\left (a x^5\right ) \left (\text {csch}^{-1}\left (a x^5\right )+2 \log \left (1-e^{-2 \text {csch}^{-1}\left (a x^5\right )}\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcsch}\left (a x^{5}\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arcsch}\left (a x^{5}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccsch}\left (a \,x^{5}\right )}{x}\, dx \]
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 \[ 5 \, a^{2} \int \frac {x^{9} \log \relax (x)}{a^{2} x^{10} + {\left (a^{2} x^{10} + 1\right )}^{\frac {3}{2}} + 1}\,{d x} - \frac {1}{2} \, \log \left (a^{2} x^{10} + 1\right ) \log \relax (x) - \log \relax (a) \log \relax (x) - \frac {5}{2} \, \log \relax (x)^{2} + \log \relax (x) \log \left (\sqrt {a^{2} x^{10} + 1} + 1\right ) - \frac {1}{20} \, {\rm Li}_2\left (-a^{2} x^{10}\right ) \]
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 \[ \int \frac {\mathrm {asinh}\left (\frac {1}{a\,x^5}\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {acsch}{\left (a x^{5} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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