Optimal. Leaf size=205 \[ -\frac{5 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^4}+\frac{5 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^4}-\frac{\sqrt{1-a^2 x^2}}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}-\frac{10 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{3 a^4} \]
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Rubi [A] time = 0.313618, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6016, 261, 5950, 5994} \[ -\frac{5 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^4}+\frac{5 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{3 a^4}-\frac{\sqrt{1-a^2 x^2}}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}-\frac{10 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{3 a^4} \]
Antiderivative was successfully verified.
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Rule 6016
Rule 261
Rule 5950
Rule 5994
Rubi steps
\begin{align*} \int \frac{x^3 \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac{2 \int \frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{3 a^2}+\frac{2 \int \frac{x^2 \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a}\\ &=-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac{\int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a^3}+\frac{4 \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a^3}+\frac{\int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{3 a^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 a^4}-\frac{x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}-\frac{10 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{3 a^4}-\frac{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac{x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac{5 i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{3 a^4}+\frac{5 i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{3 a^4}\\ \end{align*}
Mathematica [A] time = 0.391785, size = 160, normalized size = 0.78 \[ \frac{\sqrt{1-a^2 x^2} \left (-\frac{5 i \left (\text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt{1-a^2 x^2}}+\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2-\frac{5 i \tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt{1-a^2 x^2}}-3 \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x)-1\right )}{3 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.258, size = 175, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+ax{\it Artanh} \left ( ax \right ) +2\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+1}{3\,{a}^{4}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{\frac{5\,i}{3}}{\it Artanh} \left ( ax \right ) }{{a}^{4}}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{5\,i}{3}}{\it Artanh} \left ( ax \right ) }{{a}^{4}}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{5\,i}{3}}}{{a}^{4}}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{5\,i}{3}}}{{a}^{4}}{\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 \frac{x^{3} \operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{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} x^{3} \operatorname{artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, 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{x^{3} \operatorname{atanh}^{2}{\left (a x \right )}}{\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{x^{3} \operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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