Optimal. Leaf size=27 \[ a \log (x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.0636175, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5723, 29} \[ a \log (x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 5723
Rule 29
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)}{x^2 \sqrt{1+a^2 x^2}} \, dx &=-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{x}+a \int \frac{1}{x} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{x}+a \log (x)\\ \end{align*}
Mathematica [A] time = 0.0364905, size = 29, normalized size = 1.07 \[ a \log (a x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 56, normalized size = 2.1 \begin{align*} -2\,a{\it Arcsinh} \left ( ax \right ) +{\frac{{\it Arcsinh} \left ( ax \right ) }{x} \left ( ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) }+a\ln \left ( \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16535, size = 34, normalized size = 1.26 \begin{align*} a \log \left (x\right ) - \frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.65206, size = 88, normalized size = 3.26 \begin{align*} \frac{a x \log \left (x\right ) - \sqrt{a^{2} x^{2} + 1} \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} \sqrt{a^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38914, size = 113, normalized size = 4.19 \begin{align*} -a{\left (\frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \frac{{\left | a \right |} \log \left ({\left | x \right |}\right )}{a^{2}}\right )}{\left | a \right |} + \frac{2 \,{\left | a \right |} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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