Optimal. Leaf size=185 \[ -3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-6 \text{PolyLog}\left (4,-e^{\tanh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac{6 a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3 \]
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Rubi [A] time = 0.40942, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6030, 6020, 4182, 2531, 6609, 2282, 6589, 5994, 5962, 191} \[ -3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-6 \text{PolyLog}\left (4,-e^{\tanh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac{6 a x}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3 \]
Antiderivative was successfully verified.
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Rule 6030
Rule 6020
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 5994
Rule 5962
Rule 191
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac{x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-(3 a) \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\operatorname{Subst}\left (\int x^3 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=\frac{6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \operatorname{Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+3 \operatorname{Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-(6 a) \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{6 a x}{\sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-6 \operatorname{Subst}\left (\int x \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{6 a x}{\sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-6 \operatorname{Subst}\left (\int \text{Li}_3\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+6 \operatorname{Subst}\left (\int \text{Li}_3\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{6 a x}{\sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-6 \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )+6 \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-\frac{6 a x}{\sqrt{1-a^2 x^2}}+\frac{6 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x) \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )-6 \text{Li}_4\left (-e^{\tanh ^{-1}(a x)}\right )+6 \text{Li}_4\left (e^{\tanh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.339043, size = 230, normalized size = 1.24 \[ \frac{1}{8} \left (24 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )+24 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+48 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )-48 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,-e^{-\tanh ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,e^{\tanh ^{-1}(a x)}\right )-\frac{48 a x}{\sqrt{1-a^2 x^2}}+\frac{8 \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{24 a x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{48 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-2 \tanh ^{-1}(a x)^4-8 \tanh ^{-1}(a x)^3 \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+8 \tanh ^{-1}(a x)^3 \log \left (1-e^{\tanh ^{-1}(a x)}\right )+\pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.287, size = 305, normalized size = 1.7 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}-3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+6\,{\it Artanh} \left ( ax \right ) -6}{2\,ax-2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}+3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+6\,{\it Artanh} \left ( ax \right ) +6}{2\,ax+2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}- \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +6\,{\it Artanh} \left ( ax \right ){\it polylog} \left ( 3,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it polylog} \left ( 4,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) + \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it Artanh} \left ( ax \right ){\it polylog} \left ( 3,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +6\,{\it polylog} \left ( 4,{\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}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} 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{artanh}\left (a x\right )^{3}}{a^{4} x^{5} - 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{atanh}^{3}{\left (a x \right )}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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