Optimal. Leaf size=205 \[ \frac{3 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \text{PolyLog}\left (4,1-\frac{2}{1-a x}\right )}{4 a^4}+\frac{3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{\tanh ^{-1}(a x)^3}{2 a^4}-\frac{3 x \tanh ^{-1}(a x)^2}{2 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{2 a^4}+\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^4}+\frac{3 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^4} \]
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Rubi [A] time = 0.469477, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5980, 5916, 5910, 5984, 5918, 2402, 2315, 5948, 6058, 6062, 6610} \[ \frac{3 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \text{PolyLog}\left (4,1-\frac{2}{1-a x}\right )}{4 a^4}+\frac{3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{\tanh ^{-1}(a x)^3}{2 a^4}-\frac{3 x \tanh ^{-1}(a x)^2}{2 a^3}-\frac{3 \tanh ^{-1}(a x)^2}{2 a^4}+\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^4}+\frac{3 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^4} \]
Antiderivative was successfully verified.
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Rule 5980
Rule 5916
Rule 5910
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 5948
Rule 6058
Rule 6062
Rule 6610
Rubi steps
\begin{align*} \int \frac{x^3 \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx &=-\frac{\int x \tanh ^{-1}(a x)^3 \, dx}{a^2}+\frac{\int \frac{x \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{\int \frac{\tanh ^{-1}(a x)^3}{1-a x} \, dx}{a^3}+\frac{3 \int \frac{x^2 \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a}\\ &=-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{\tanh ^{-1}(a x)^3 \log \left (\frac{2}{1-a x}\right )}{a^4}-\frac{3 \int \tanh ^{-1}(a x)^2 \, dx}{2 a^3}+\frac{3 \int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a^3}-\frac{3 \int \frac{\tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}\\ &=-\frac{3 x \tanh ^{-1}(a x)^2}{2 a^3}+\frac{\tanh ^{-1}(a x)^3}{2 a^4}-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{\tanh ^{-1}(a x)^3 \log \left (\frac{2}{1-a x}\right )}{a^4}+\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 \int \frac{\tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}+\frac{3 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac{3 \tanh ^{-1}(a x)^2}{2 a^4}-\frac{3 x \tanh ^{-1}(a x)^2}{2 a^3}+\frac{\tanh ^{-1}(a x)^3}{2 a^4}-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{\tanh ^{-1}(a x)^3 \log \left (\frac{2}{1-a x}\right )}{a^4}+\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 \tanh ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \int \frac{\text{Li}_3\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{2 a^3}+\frac{3 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{a^3}\\ &=-\frac{3 \tanh ^{-1}(a x)^2}{2 a^4}-\frac{3 x \tanh ^{-1}(a x)^2}{2 a^3}+\frac{\tanh ^{-1}(a x)^3}{2 a^4}-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{3 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{a^4}+\frac{\tanh ^{-1}(a x)^3 \log \left (\frac{2}{1-a x}\right )}{a^4}+\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 \tanh ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \text{Li}_4\left (1-\frac{2}{1-a x}\right )}{4 a^4}-\frac{3 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}\\ &=-\frac{3 \tanh ^{-1}(a x)^2}{2 a^4}-\frac{3 x \tanh ^{-1}(a x)^2}{2 a^3}+\frac{\tanh ^{-1}(a x)^3}{2 a^4}-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{3 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{a^4}+\frac{\tanh ^{-1}(a x)^3 \log \left (\frac{2}{1-a x}\right )}{a^4}+\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 \tanh ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \text{Li}_4\left (1-\frac{2}{1-a x}\right )}{4 a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{a^4}\\ &=-\frac{3 \tanh ^{-1}(a x)^2}{2 a^4}-\frac{3 x \tanh ^{-1}(a x)^2}{2 a^3}+\frac{\tanh ^{-1}(a x)^3}{2 a^4}-\frac{x^2 \tanh ^{-1}(a x)^3}{2 a^2}-\frac{\tanh ^{-1}(a x)^4}{4 a^4}+\frac{3 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{a^4}+\frac{\tanh ^{-1}(a x)^3 \log \left (\frac{2}{1-a x}\right )}{a^4}+\frac{3 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \tanh ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 \tanh ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \text{Li}_4\left (1-\frac{2}{1-a x}\right )}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.2687, size = 142, normalized size = 0.69 \[ -\frac{6 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+6 \left (\tanh ^{-1}(a x)^2+1\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+3 \text{PolyLog}\left (4,-e^{-2 \tanh ^{-1}(a x)}\right )-2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^4+6 a x \tanh ^{-1}(a x)^2-6 \tanh ^{-1}(a x)^2-4 \tanh ^{-1}(a x)^3 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-12 \tanh ^{-1}(a x) \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{4 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.895, size = 248, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}{4\,{a}^{4}}}-{\frac{{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{2\,{a}^{2}}}-{\frac{3\,x \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2\,{a}^{3}}}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{2\,{a}^{4}}}-{\frac{3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2\,{a}^{4}}}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}{{a}^{4}}\ln \left ({\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}}+1 \right ) }+{\frac{3\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2\,{a}^{4}}{\it polylog} \left ( 2,-{\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}} \right ) }-{\frac{3\,{\it Artanh} \left ( ax \right ) }{2\,{a}^{4}}{\it polylog} \left ( 3,-{\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}} \right ) }+{\frac{3}{4\,{a}^{4}}{\it polylog} \left ( 4,-{\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}} \right ) }+3\,{\frac{{\it Artanh} \left ( ax \right ) }{{a}^{4}}\ln \left ({\frac{ \left ( ax+1 \right ) ^{2}}{-{a}^{2}{x}^{2}+1}}+1 \right ) }+{\frac{3}{2\,{a}^{4}}{\it polylog} \left ( 2,-{\frac{ \left ( ax+1 \right ) ^{2}}{-{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*} \frac{4 \,{\left (a^{2} x^{2} + \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{3} + \log \left (-a x + 1\right )^{4}}{64 \, a^{4}} - \frac{1}{8} \, \int \frac{2 \, a^{3} x^{3} \log \left (a x + 1\right )^{3} - 6 \, a^{3} x^{3} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \,{\left (a^{3} x^{3} + a^{2} x^{2} +{\left (2 \, a^{3} x^{3} + a x + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{2 \,{\left (a^{5} x^{2} - a^{3}\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 (-\frac{x^{3} \operatorname{artanh}\left (a x\right )^{3}}{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}^{3}{\left (a x \right )}}{a^{2} x^{2} - 1}\, 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 )^{3}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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