Optimal. Leaf size=243 \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{16 a^3}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{16 a^3}-\frac{\left (1-a^2 x^2\right )^{5/2}}{30 a^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{72 a^3}+\frac{\sqrt{1-a^2 x^2}}{16 a^3}-\frac{1}{6} a^2 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{7}{24} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{8 a^3} \]
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Rubi [A] time = 0.572297, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6014, 6010, 6016, 261, 5950, 266, 43} \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{16 a^3}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{16 a^3}-\frac{\left (1-a^2 x^2\right )^{5/2}}{30 a^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{72 a^3}+\frac{\sqrt{1-a^2 x^2}}{16 a^3}-\frac{1}{6} a^2 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{7}{24} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 6014
Rule 6010
Rule 6016
Rule 261
Rule 5950
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x) \, dx &=-\left (a^2 \int x^4 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx\right )+\int x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx\\ &=\frac{1}{4} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{1}{4} \int \frac{x^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx-\frac{1}{4} a \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx-\frac{1}{6} a^2 \int \frac{x^4 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx+\frac{1}{6} a^3 \int \frac{x^5}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2}+\frac{7}{24} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{8} \int \frac{x^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx+\frac{\int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{8 a^2}+\frac{\int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{8 a}-\frac{1}{24} a \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx-\frac{1}{8} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )+\frac{1}{12} a^3 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{8 a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a^2}+\frac{7}{24} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{\tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{4 a^3}-\frac{i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{8 a^3}+\frac{i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{8 a^3}-\frac{\int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{16 a^2}-\frac{\int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{16 a}-\frac{1}{48} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )-\frac{1}{8} a \operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )+\frac{1}{12} a^3 \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{1-a^2 x}}-\frac{2 \sqrt{1-a^2 x}}{a^4}+\frac{\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )\\ &=\frac{\sqrt{1-a^2 x^2}}{48 a^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{36 a^3}-\frac{\left (1-a^2 x^2\right )^{5/2}}{30 a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a^2}+\frac{7}{24} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{\tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{8 a^3}-\frac{i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{16 a^3}+\frac{i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{16 a^3}-\frac{1}{48} a \operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac{\sqrt{1-a^2 x^2}}{16 a^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{72 a^3}-\frac{\left (1-a^2 x^2\right )^{5/2}}{30 a^3}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{16 a^2}+\frac{7}{24} x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{1}{6} a^2 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{\tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{8 a^3}-\frac{i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{16 a^3}+\frac{i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{16 a^3}\\ \end{align*}
Mathematica [A] time = 0.906854, size = 224, normalized size = 0.92 \[ \frac{-45 i \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )+45 i \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )-24 a^4 x^4 \sqrt{1-a^2 x^2}+38 a^2 x^2 \sqrt{1-a^2 x^2}+31 \sqrt{1-a^2 x^2}-120 a^5 x^5 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+210 a^3 x^3 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-45 a x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-45 i \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )+45 i \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )}{720 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.195, size = 195, normalized size = 0.8 \begin{align*} -{\frac{120\,{\it Artanh} \left ( ax \right ){x}^{5}{a}^{5}+24\,{x}^{4}{a}^{4}-210\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) -38\,{a}^{2}{x}^{2}+45\,ax{\it Artanh} \left ( ax \right ) -31}{720\,{a}^{3}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{\frac{i}{16}}{\it Artanh} \left ( ax \right ) }{{a}^{3}}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{i}{16}}{\it Artanh} \left ( ax \right ) }{{a}^{3}}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{i}{16}}}{{a}^{3}}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{i}{16}}}{{a}^{3}}{\it dilog} \left ( 1-{i \left ( ax+1 \right ){\frac{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{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2} \operatorname{artanh}\left (a x\right )\,{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 (-{\left (a^{2} x^{4} - x^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right ), 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 x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \operatorname{atanh}{\left (a x \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{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2} \operatorname{artanh}\left (a x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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