Optimal. Leaf size=77 \[ \frac{3 a^3 \sqrt{a^2 x^2+1}}{40 x^2}-\frac{a \sqrt{a^2 x^2+1}}{20 x^4}-\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{5 x^5} \]
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Rubi [A] time = 0.0449983, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5661, 266, 51, 63, 208} \[ \frac{3 a^3 \sqrt{a^2 x^2+1}}{40 x^2}-\frac{a \sqrt{a^2 x^2+1}}{20 x^4}-\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)}{x^6} \, dx &=-\frac{\sinh ^{-1}(a x)}{5 x^5}+\frac{1}{5} a \int \frac{1}{x^5 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sinh ^{-1}(a x)}{5 x^5}+\frac{1}{10} a \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a \sqrt{1+a^2 x^2}}{20 x^4}-\frac{\sinh ^{-1}(a x)}{5 x^5}-\frac{1}{40} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a \sqrt{1+a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1+a^2 x^2}}{40 x^2}-\frac{\sinh ^{-1}(a x)}{5 x^5}+\frac{1}{80} \left (3 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a \sqrt{1+a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1+a^2 x^2}}{40 x^2}-\frac{\sinh ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2}} \, dx,x,\sqrt{1+a^2 x^2}\right )\\ &=-\frac{a \sqrt{1+a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1+a^2 x^2}}{40 x^2}-\frac{\sinh ^{-1}(a x)}{5 x^5}-\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [C] time = 0.0111742, size = 49, normalized size = 0.64 \[ -\frac{1}{5} a^5 \sqrt{a^2 x^2+1} \text{Hypergeometric2F1}\left (\frac{1}{2},3,\frac{3}{2},a^2 x^2+1\right )-\frac{\sinh ^{-1}(a x)}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 70, normalized size = 0.9 \begin{align*}{a}^{5} \left ( -{\frac{{\it Arcsinh} \left ( ax \right ) }{5\,{a}^{5}{x}^{5}}}-{\frac{1}{20\,{a}^{4}{x}^{4}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{3}{40\,{a}^{2}{x}^{2}}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{3}{40}{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16405, size = 88, normalized size = 1.14 \begin{align*} -\frac{1}{40} \,{\left (3 \, a^{4} \operatorname{arsinh}\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{arsinh}\left (a x\right )}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95841, size = 302, normalized size = 3.92 \begin{align*} -\frac{3 \, a^{5} x^{5} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{5} x^{5} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 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{asinh}{\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 = 1.42769, size = 128, normalized size = 1.66 \begin{align*} \frac{1}{80} \, a^{5}{\left (\frac{2 \,{\left (3 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{a^{2} x^{2} + 1}\right )}}{a^{4} x^{4}} - 3 \, \log \left (\sqrt{a^{2} x^{2} + 1} + 1\right ) + 3 \, \log \left (\sqrt{a^{2} x^{2} + 1} - 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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