Optimal. Leaf size=102 \[ -3 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+3 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+6 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )-6 \sinh ^{-1}(a x) \text{PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-6 \text{PolyLog}\left (4,-e^{\sinh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.162071, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5760, 4182, 2531, 6609, 2282, 6589} \[ -3 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+3 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+6 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )-6 \sinh ^{-1}(a x) \text{PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-6 \text{PolyLog}\left (4,-e^{\sinh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 5760
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^3}{x \sqrt{1+a^2 x^2}} \, dx &=\operatorname{Subst}\left (\int x^3 \text{csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-3 \operatorname{Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+3 \operatorname{Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-3 \sinh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+3 \sinh ^{-1}(a x)^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+6 \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-6 \operatorname{Subst}\left (\int x \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-3 \sinh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+3 \sinh ^{-1}(a x)^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+6 \sinh ^{-1}(a x) \text{Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-6 \sinh ^{-1}(a x) \text{Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-6 \operatorname{Subst}\left (\int \text{Li}_3\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+6 \operatorname{Subst}\left (\int \text{Li}_3\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-3 \sinh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+3 \sinh ^{-1}(a x)^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+6 \sinh ^{-1}(a x) \text{Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-6 \sinh ^{-1}(a x) \text{Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-6 \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+6 \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-3 \sinh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+3 \sinh ^{-1}(a x)^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )+6 \sinh ^{-1}(a x) \text{Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )-6 \sinh ^{-1}(a x) \text{Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-6 \text{Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )+6 \text{Li}_4\left (e^{\sinh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.128932, size = 146, normalized size = 1.43 \[ \frac{1}{8} \left (24 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )+24 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+48 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(a x)}\right )-48 \sinh ^{-1}(a x) \text{PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,-e^{-\sinh ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x)^4-8 \sinh ^{-1}(a x)^3 \log \left (e^{-\sinh ^{-1}(a x)}+1\right )+8 \sinh ^{-1}(a x)^3 \log \left (1-e^{\sinh ^{-1}(a x)}\right )+\pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.056, size = 197, normalized size = 1.9 \begin{align*} - \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}\ln \left ( 1+ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) +6\,{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 3,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) -6\,{\it polylog} \left ( 4,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) + \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}\ln \left ( 1-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) +3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -6\,{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 3,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) +6\,{\it polylog} \left ( 4,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 )^{3}}{\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 )^{3}}{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}^{3}{\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 )^{3}}{\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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