Optimal. Leaf size=54 \[ \frac{1}{10} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (a x^5\right )}\right )-\frac{1}{10} \cosh ^{-1}\left (a x^5\right )^2+\frac{1}{5} \cosh ^{-1}\left (a x^5\right ) \log \left (e^{2 \cosh ^{-1}\left (a x^5\right )}+1\right ) \]
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Rubi [A] time = 0.0668968, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5891, 3718, 2190, 2279, 2391} \[ \frac{1}{10} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (a x^5\right )}\right )-\frac{1}{10} \cosh ^{-1}\left (a x^5\right )^2+\frac{1}{5} \cosh ^{-1}\left (a x^5\right ) \log \left (e^{2 \cosh ^{-1}\left (a x^5\right )}+1\right ) \]
Antiderivative was successfully verified.
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Rule 5891
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}\left (a x^5\right )}{x} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}\left (a x^5\right )\right )\\ &=-\frac{1}{10} \cosh ^{-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,\cosh ^{-1}\left (a x^5\right )\right )\\ &=-\frac{1}{10} \cosh ^{-1}\left (a x^5\right )^2+\frac{1}{5} \cosh ^{-1}\left (a x^5\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^5\right )}\right )-\frac{1}{5} \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (a x^5\right )\right )\\ &=-\frac{1}{10} \cosh ^{-1}\left (a x^5\right )^2+\frac{1}{5} \cosh ^{-1}\left (a x^5\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^5\right )}\right )-\frac{1}{10} \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (a x^5\right )}\right )\\ &=-\frac{1}{10} \cosh ^{-1}\left (a x^5\right )^2+\frac{1}{5} \cosh ^{-1}\left (a x^5\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^5\right )}\right )+\frac{1}{10} \text{Li}_2\left (-e^{2 \cosh ^{-1}\left (a x^5\right )}\right )\\ \end{align*}
Mathematica [A] time = 0.0418792, size = 50, normalized size = 0.93 \[ \frac{1}{10} \left (\cosh ^{-1}\left (a x^5\right ) \left (\cosh ^{-1}\left (a x^5\right )+2 \log \left (e^{-2 \cosh ^{-1}\left (a x^5\right )}+1\right )\right )-\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}\left (a x^5\right )}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccosh} \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*} \int \frac{\operatorname{arcosh}\left (a x^{5}\right )}{x}\,{d x} \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{arcosh}\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{acosh}{\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{arcosh}\left (a x^{5}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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