Optimal. Leaf size=54 \[ \frac{1}{4} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (a x^2\right )}\right )-\frac{1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac{1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right ) \]
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Rubi [A] time = 0.0622429, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5890, 3716, 2190, 2279, 2391} \[ \frac{1}{4} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (a x^2\right )}\right )-\frac{1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac{1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 5890
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}\left (a x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (a x^2\right )\right )\\ &=-\frac{1}{4} \sinh ^{-1}\left (a x^2\right )^2-\operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (a x^2\right )\right )\\ &=-\frac{1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac{1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (a x^2\right )\right )\\ &=-\frac{1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac{1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (a x^2\right )}\right )\\ &=-\frac{1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac{1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right )+\frac{1}{4} \text{Li}_2\left (e^{2 \sinh ^{-1}\left (a x^2\right )}\right )\\ \end{align*}
Mathematica [A] time = 0.0070841, size = 54, normalized size = 1. \[ \frac{1}{4} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (a x^2\right )}\right )-\frac{1}{4} \sinh ^{-1}\left (a x^2\right )^2+\frac{1}{2} \sinh ^{-1}\left (a x^2\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^2\right )}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Arcsinh} \left ( a{x}^{2} \right ) }{x}}\, dx \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*} \int \frac{\operatorname{arsinh}\left (a x^{2}\right )}{x}\,{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{\operatorname{arsinh}\left (a x^{2}\right )}{x}, 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{\operatorname{asinh}{\left (a x^{2} \right )}}{x}\, 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{\operatorname{arsinh}\left (a x^{2}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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