Optimal. Leaf size=68 \[ -2 \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.148574, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5760, 4182, 2531, 2282, 6589} \[ -2 \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 5760
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx &=\operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 \operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+2 \operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+2 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-2 \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 \sinh ^{-1}(a x) \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+2 \text{Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-2 \text{Li}_3\left (e^{\sinh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.11136, size = 100, normalized size = 1.47 \[ 2 \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \text{PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,e^{-\sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x)^2 \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x)^2 \log \left (e^{-\sinh ^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.052, size = 144, normalized size = 2.1 \begin{align*} - \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1+ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -2\,{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 2,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) +2\,{\it polylog} \left ( 3,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) + \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) +2\,{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 2,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -2\,{\it polylog} \left ( 3,ax+\sqrt{{a}^{2}{x}^{2}+1} \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*} \int \frac{\operatorname{arsinh}\left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1} 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{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )^{2}}{a^{2} x^{3} + 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}^{2}{\left (a x \right )}}{x \sqrt{a^{2} x^{2} + 1}}\, 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\right )^{2}}{\sqrt{a^{2} x^{2} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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