Optimal. Leaf size=54 \[ -\frac{a \sqrt{a^2 x^2+1}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{3 x^3} \]
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Rubi [A] time = 0.032432, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -\frac{a \sqrt{a^2 x^2+1}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{3 x^3} \]
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^4} \, dx &=-\frac{\sinh ^{-1}(a x)}{3 x^3}+\frac{1}{3} a \int \frac{1}{x^3 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sinh ^{-1}(a x)}{3 x^3}+\frac{1}{6} a \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}}{6 x^2}-\frac{\sinh ^{-1}(a x)}{3 x^3}-\frac{1}{12} a^3 \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}}{6 x^2}-\frac{\sinh ^{-1}(a x)}{3 x^3}-\frac{1}{6} a \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}}{6 x^2}-\frac{\sinh ^{-1}(a x)}{3 x^3}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.01014, size = 54, normalized size = 1. \[ -\frac{a \sqrt{a^2 x^2+1}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 51, normalized size = 0.9 \begin{align*}{a}^{3} \left ( -{\frac{{\it Arcsinh} \left ( ax \right ) }{3\,{a}^{3}{x}^{3}}}-{\frac{1}{6\,{a}^{2}{x}^{2}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{1}{6}{\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 = 1.11922, size = 61, normalized size = 1.13 \begin{align*} \frac{1}{6} \,{\left (a^{2} \operatorname{arsinh}\left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) - \frac{\sqrt{a^{2} x^{2} + 1}}{x^{2}}\right )} a - \frac{\operatorname{arsinh}\left (a x\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03025, size = 273, normalized size = 5.06 \begin{align*} \frac{a^{3} x^{3} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - a^{3} x^{3} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) + 2 \, x^{3} \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right ) - \sqrt{a^{2} x^{2} + 1} a x + 2 \,{\left (x^{3} - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{6 \, x^{3}} \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^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37784, size = 104, normalized size = 1.93 \begin{align*} -\frac{1}{12} \, a^{3}{\left (\frac{2 \, \sqrt{a^{2} x^{2} + 1}}{a^{2} x^{2}} - \log \left (\sqrt{a^{2} x^{2} + 1} + 1\right ) + \log \left (\sqrt{a^{2} x^{2} + 1} - 1\right )\right )} - \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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