Optimal. Leaf size=27 \[ -a \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.0214737, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5661, 266, 63, 208} \[ -a \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)}{x^2} \, dx &=-\frac{\sinh ^{-1}(a x)}{x}+a \int \frac{1}{x \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sinh ^{-1}(a x)}{x}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sinh ^{-1}(a x)}{x}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2}} \, dx,x,\sqrt{1+a^2 x^2}\right )}{a}\\ &=-\frac{\sinh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0021349, size = 27, normalized size = 1. \[ -a \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-\frac{\sinh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 30, normalized size = 1.1 \begin{align*} a \left ( -{\frac{{\it Arcsinh} \left ( ax \right ) }{ax}}-{\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 = 1.12306, size = 32, normalized size = 1.19 \begin{align*} -a \operatorname{arsinh}\left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) - \frac{\operatorname{arsinh}\left (a x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94552, size = 212, normalized size = 7.85 \begin{align*} -\frac{a x \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - a x \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) -{\left (x - 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - x \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right )}{x} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38967, size = 76, normalized size = 2.81 \begin{align*} -\frac{1}{2} \, a{\left (\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 )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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