Optimal. Leaf size=103 \[ -\frac{3 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^3}+\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a^3}+\frac{\tanh ^{-1}(a x)^4}{4 a^3}-\frac{x \tanh ^{-1}(a x)^3}{a^2}-\frac{\tanh ^{-1}(a x)^3}{a^3}+\frac{3 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3} \]
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Rubi [A] time = 0.269013, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {5980, 5910, 5984, 5918, 5948, 6058, 6610} \[ -\frac{3 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^3}+\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a^3}+\frac{\tanh ^{-1}(a x)^4}{4 a^3}-\frac{x \tanh ^{-1}(a x)^3}{a^2}-\frac{\tanh ^{-1}(a x)^3}{a^3}+\frac{3 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3} \]
Antiderivative was successfully verified.
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Rule 5980
Rule 5910
Rule 5984
Rule 5918
Rule 5948
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int \frac{x^2 \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx &=-\frac{\int \tanh ^{-1}(a x)^3 \, dx}{a^2}+\frac{\int \frac{\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac{x \tanh ^{-1}(a x)^3}{a^2}+\frac{\tanh ^{-1}(a x)^4}{4 a^3}+\frac{3 \int \frac{x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a}\\ &=-\frac{\tanh ^{-1}(a x)^3}{a^3}-\frac{x \tanh ^{-1}(a x)^3}{a^2}+\frac{\tanh ^{-1}(a x)^4}{4 a^3}+\frac{3 \int \frac{\tanh ^{-1}(a x)^2}{1-a x} \, dx}{a^2}\\ &=-\frac{\tanh ^{-1}(a x)^3}{a^3}-\frac{x \tanh ^{-1}(a x)^3}{a^2}+\frac{\tanh ^{-1}(a x)^4}{4 a^3}+\frac{3 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^3}-\frac{6 \int \frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac{\tanh ^{-1}(a x)^3}{a^3}-\frac{x \tanh ^{-1}(a x)^3}{a^2}+\frac{\tanh ^{-1}(a x)^4}{4 a^3}+\frac{3 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^3}+\frac{3 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a^3}-\frac{3 \int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}\\ &=-\frac{\tanh ^{-1}(a x)^3}{a^3}-\frac{x \tanh ^{-1}(a x)^3}{a^2}+\frac{\tanh ^{-1}(a x)^4}{4 a^3}+\frac{3 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^3}+\frac{3 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a^3}-\frac{3 \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.242113, size = 78, normalized size = 0.76 \[ \frac{-12 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-6 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x)^2 \left (\tanh ^{-1}(a x)^2+(4-4 a x) \tanh ^{-1}(a x)+12 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right )}{4 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.327, size = 788, normalized size = 7.7 \begin{align*} \text{result too large to display} \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 (2 \, a x - \log \left (a x + 1\right ) - 2\right )} \log \left (-a x + 1\right )^{3} + \log \left (-a x + 1\right )^{4} - 6 \,{\left (4 \,{\left (a x + 1\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2}\right )} \log \left (-a x + 1\right )^{2}}{64 \, a^{3}} + \frac{1}{8} \, \int -\frac{2 \, a^{2} x^{2} \log \left (a x + 1\right )^{3} - 3 \,{\left ({\left (2 \, a^{2} x^{2} - a x - 1\right )} \log \left (a x + 1\right )^{2} + 4 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{2 \,{\left (a^{4} x^{2} - a^{2}\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^{2} \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^{2} \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^{2} \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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