Optimal. Leaf size=187 \[ -6 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+6 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+6 a \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-6 a \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-\frac{6 a}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-6 a \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]
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Rubi [A] time = 0.424716, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6030, 6008, 6020, 4182, 2531, 2282, 6589, 5962, 5958} \[ -6 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+6 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+6 a \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-6 a \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-\frac{6 a}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-6 a \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 6030
Rule 6008
Rule 6020
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 5962
Rule 5958
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}+(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx+\left (6 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{6 a}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}+(3 a) \operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{6 a}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-6 a \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}-(6 a) \operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+(6 a) \operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{6 a}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-6 a \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}-6 a \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+6 a \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+(6 a) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-(6 a) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{6 a}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-6 a \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}-6 a \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+6 a \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+(6 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )-(6 a) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-\frac{6 a}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-6 a \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3}{x}-6 a \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+6 a \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+6 a \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-6 a \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 1.88996, size = 270, normalized size = 1.44 \[ 6 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-6 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+6 a \text{PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )-6 a \text{PolyLog}\left (3,e^{-\tanh ^{-1}(a x)}\right )-\frac{6 a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{6 a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^3 \sinh ^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{x}-\frac{a^2 x \tanh ^{-1}(a x)^3 \text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{4 \sqrt{1-a^2 x^2}}+3 a \tanh ^{-1}(a x)^2 \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-3 a \tanh ^{-1}(a x)^2 \log \left (e^{-\tanh ^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.292, size = 282, normalized size = 1.5 \begin{align*} -{\frac{a \left ( \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}-3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+6\,{\it Artanh} \left ( ax \right ) -6 \right ) }{2\,ax-2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{ \left ( \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}+3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+6\,{\it Artanh} \left ( ax \right ) +6 \right ) a}{2\,ax+2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{x}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-3\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -6\,a{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +6\,a{\it polylog} \left ( 3,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +6\,a{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -6\,a{\it polylog} \left ( 3,{\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^{2}}\,{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^{6} - 2 \, a^{2} x^{4} + x^{2}}, 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^{2} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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