Optimal. Leaf size=221 \[ -3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )+\frac{2 a^2}{\sqrt{1-a^2 x^2}}-\frac{2 a^3 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\frac{a^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]
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Rubi [A] time = 0.776324, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {6030, 6026, 6008, 266, 63, 208, 6020, 4182, 2531, 2282, 6589, 5994, 5958} \[ -3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )+\frac{2 a^2}{\sqrt{1-a^2 x^2}}-\frac{2 a^3 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}+\frac{a^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 6030
Rule 6026
Rule 6008
Rule 266
Rule 63
Rule 208
Rule 6020
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 5994
Rule 5958
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a \int \frac{\tanh ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{2} a^2 \int \frac{\tanh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx+a^2 \int \frac{\tanh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx+a^4 \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+\frac{a^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )+a^2 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx+a^2 \operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )-\left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 a^2}{\sqrt{1-a^2 x^2}}-\frac{2 a^3 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+\frac{a^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )-a^2 \operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+a^2 \operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\left (2 a^2\right ) \operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\left (2 a^2\right ) \operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=\frac{2 a^2}{\sqrt{1-a^2 x^2}}-\frac{2 a^3 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+\frac{a^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-a^2 \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+\left (2 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\left (2 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=\frac{2 a^2}{\sqrt{1-a^2 x^2}}-\frac{2 a^3 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+\frac{a^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )-a^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )+\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )-\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=\frac{2 a^2}{\sqrt{1-a^2 x^2}}-\frac{2 a^3 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+\frac{a^2 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-3 a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+3 a^2 \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-3 a^2 \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 2.7034, size = 266, normalized size = 1.2 \[ \frac{1}{8} a^2 \left (24 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-24 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+24 \text{PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )-24 \text{PolyLog}\left (3,e^{-\tanh ^{-1}(a x)}\right )+\frac{16}{\sqrt{1-a^2 x^2}}+\frac{8 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{16 a x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{2 a x \tanh ^{-1}(a x) \text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+4 \tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \tanh ^{-1}(a x)+12 \tanh ^{-1}(a x)^2 \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-12 \tanh ^{-1}(a x)^2 \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+8 \log \left (\tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )+\tanh ^{-1}(a x)^2 \left (-\text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )-\tanh ^{-1}(a x)^2 \text{sech}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.299, size = 313, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2} \left ( \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-2\,{\it Artanh} \left ( ax \right ) +2 \right ) }{2\,ax-2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{ \left ( \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+2\,{\it Artanh} \left ( ax \right ) +2 \right ){a}^{2}}{2\,ax+2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{\it Artanh} \left ( ax \right ) \left ( 2\,ax+{\it Artanh} \left ( ax \right ) \right ) }{2\,{x}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-2\,{a}^{2}{\it Artanh} \left ({\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -{\frac{3\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-3\,{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +3\,{a}^{2}{\it polylog} \left ( 3,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +{\frac{3\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+3\,{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -3\,{a}^{2}{\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 )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} 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 )^{2}}{a^{4} x^{7} - 2 \, 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}^{2}{\left (a x \right )}}{x^{3} \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 )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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