Optimal. Leaf size=37 \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.0246061, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5665, 3301} \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5665
Rule 3301
Rubi steps
\begin{align*} \int \frac{x}{\sinh ^{-1}(a x)^2} \, dx &=-\frac{x \sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x \sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0032678, size = 32, normalized size = 0.86 \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac{\sinh \left (2 \sinh ^{-1}(a x)\right )}{2 a^2 \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 28, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sinh \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{2\,{\it Arcsinh} \left ( ax \right ) }}+{\it Chi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} x^{4} + a x^{2} +{\left (a^{2} x^{3} + x\right )} \sqrt{a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )} + \int \frac{2 \, a^{5} x^{5} + 2 \,{\left (a^{2} x^{2} + 1\right )} a^{3} x^{3} + 4 \, a^{3} x^{3} + 2 \, a x +{\left (4 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 1\right )} \sqrt{a^{2} x^{2} + 1}}{{\left (a^{5} x^{4} +{\left (a^{2} x^{2} + 1\right )} a^{3} x^{2} + 2 \, a^{3} x^{2} + 2 \,{\left (a^{4} x^{3} + a^{2} x\right )} \sqrt{a^{2} x^{2} + 1} + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{\operatorname{arsinh}\left (a x\right )^{2}}, x\right ) \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{x}{\operatorname{asinh}^{2}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\operatorname{arsinh}\left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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