Optimal. Leaf size=267 \[ 3 a^2 \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-3 a^2 \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text{PolyLog}\left (4,-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.447053, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6026, 6008, 6018, 6020, 4182, 2531, 6609, 2282, 6589} \[ 3 a^2 \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-3 a^2 \text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text{PolyLog}\left (4,-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6026
Rule 6008
Rule 6018
Rule 6020
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} (3 a) \int \frac{\tanh ^{-1}(a x)^2}{x^2 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{2} a^2 \int \frac{\tanh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int x^3 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )+\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-\frac{3 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-6 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-3 a^2 \text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text{Li}_4\left (-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_4\left (e^{\tanh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 8.7494, size = 416, normalized size = 1.56 \[ \frac{1}{16} a^2 \tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \left (24 \tanh ^{-1}(a x)^2 \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+48 \tanh ^{-1}(a x) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \text{PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )-48 \tanh ^{-1}(a x) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )+24 \left (\tanh ^{-1}(a x)^2+2\right ) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-48 \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+48 \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \text{PolyLog}\left (4,-e^{-\tanh ^{-1}(a x)}\right )+48 \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac{4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{a x}-\frac{a x \tanh ^{-1}(a x)^3 \text{csch}^4\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+12 \tanh ^{-1}(a x)^2-12 \tanh ^{-1}(a x)^2 \coth ^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )-2 \tanh ^{-1}(a x)^4 \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )+\pi ^4 \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )-8 \tanh ^{-1}(a x)^3 \log \left (e^{-\tanh ^{-1}(a x)}+1\right ) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )+8 \tanh ^{-1}(a x)^3 \log \left (1-e^{\tanh ^{-1}(a x)}\right ) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )+48 \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right ) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )-48 \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right ) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.321, size = 386, normalized size = 1.5 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2} \left ( 3\,ax+{\it Artanh} \left ( ax \right ) \right ) }{2\,{x}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{2}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{3\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}{\it polylog} \left ( 2,-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+3\,{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 3,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -3\,{a}^{2}{\it polylog} \left ( 4,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{2}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{3\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}{\it polylog} \left ( 2,{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-3\,{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 3,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,{a}^{2}{\it polylog} \left ( 4,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -3\,{a}^{2}{\it Artanh} \left ( ax \right ) \ln \left ( 1+{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -3\,{a}^{2}{\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,{a}^{2}{\it Artanh} \left ( ax \right ) \ln \left ( 1-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,{a}^{2}{\it polylog} \left ( 2,{\frac{ax+1}{\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{artanh}\left (a x\right )^{3}}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{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{artanh}\left (a x\right )^{3}}{a^{2} x^{5} - x^{3}}, 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{atanh}^{3}{\left (a x \right )}}{x^{3} \sqrt{- \left (a x - 1\right ) \left (a x + 1\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{\operatorname{artanh}\left (a x\right )^{3}}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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