Optimal. Leaf size=93 \[ \frac{3 a^3 \sqrt{a x-1} \sqrt{a x+1}}{40 x^2}+\frac{3}{40} a^5 \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1}}{20 x^4}-\frac{\cosh ^{-1}(a x)}{5 x^5} \]
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Rubi [A] time = 0.0407888, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5662, 103, 12, 92, 205} \[ \frac{3 a^3 \sqrt{a x-1} \sqrt{a x+1}}{40 x^2}+\frac{3}{40} a^5 \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1}}{20 x^4}-\frac{\cosh ^{-1}(a x)}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 5662
Rule 103
Rule 12
Rule 92
Rule 205
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)}{x^6} \, dx &=-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{5} a \int \frac{1}{x^5 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{20} a \int \frac{3 a^2}{x^3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{20} \left (3 a^3\right ) \int \frac{1}{x^3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}+\frac{3 a^3 \sqrt{-1+a x} \sqrt{1+a x}}{40 x^2}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^3\right ) \int \frac{a^2}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}+\frac{3 a^3 \sqrt{-1+a x} \sqrt{1+a x}}{40 x^2}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^5\right ) \int \frac{1}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}+\frac{3 a^3 \sqrt{-1+a x} \sqrt{1+a x}}{40 x^2}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^6\right ) \operatorname{Subst}\left (\int \frac{1}{a+a x^2} \, dx,x,\sqrt{-1+a x} \sqrt{1+a x}\right )\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{20 x^4}+\frac{3 a^3 \sqrt{-1+a x} \sqrt{1+a x}}{40 x^2}-\frac{\cosh ^{-1}(a x)}{5 x^5}+\frac{3}{40} a^5 \tan ^{-1}\left (\sqrt{-1+a x} \sqrt{1+a x}\right )\\ \end{align*}
Mathematica [A] time = 0.0436838, size = 104, normalized size = 1.12 \[ -\frac{-3 a^5 x^5+a^3 x^3-3 a^5 x^5 \sqrt{a^2 x^2-1} \tan ^{-1}\left (\sqrt{a^2 x^2-1}\right )+2 a x+8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{40 x^5 \sqrt{a x-1} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 95, normalized size = 1. \begin{align*} -{\frac{{\rm arccosh} \left (ax\right )}{5\,{x}^{5}}}-{\frac{3\,{a}^{5}}{40}\sqrt{ax-1}\sqrt{ax+1}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}}}+{\frac{3\,{a}^{3}}{40\,{x}^{2}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{a}{20\,{x}^{4}}\sqrt{ax-1}\sqrt{ax+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76618, size = 88, normalized size = 0.95 \begin{align*} -\frac{1}{40} \,{\left (3 \, a^{4} \arcsin \left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) - \frac{3 \, \sqrt{a^{2} x^{2} - 1} a^{2}}{x^{2}} - \frac{2 \, \sqrt{a^{2} x^{2} - 1}}{x^{4}}\right )} a - \frac{\operatorname{arcosh}\left (a x\right )}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51877, size = 238, normalized size = 2.56 \begin{align*} \frac{6 \, a^{5} x^{5} \arctan \left (-a x + \sqrt{a^{2} x^{2} - 1}\right ) + 8 \, x^{5} \log \left (-a x + \sqrt{a^{2} x^{2} - 1}\right ) + 8 \,{\left (x^{5} - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) +{\left (3 \, a^{3} x^{3} + 2 \, a x\right )} \sqrt{a^{2} x^{2} - 1}}{40 \, x^{5}} \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 \right )}}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3486, size = 103, normalized size = 1.11 \begin{align*} \frac{1}{40} \, a^{5}{\left (\frac{3 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{a^{2} x^{2} - 1}}{a^{4} x^{4}} + 3 \, \arctan \left (\sqrt{a^{2} x^{2} - 1}\right )\right )} - \frac{\log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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