Optimal. Leaf size=78 \[ -\frac{\text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^2}+\frac{\tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a^2}-\frac{\tanh ^{-1}(a x)^3}{3 a^2}+\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^2} \]
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Rubi [A] time = 0.157278, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5984, 5918, 5948, 6058, 6610} \[ -\frac{\text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^2}+\frac{\tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{a^2}-\frac{\tanh ^{-1}(a x)^3}{3 a^2}+\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^2} \]
Antiderivative was successfully verified.
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Rule 5984
Rule 5918
Rule 5948
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int \frac{x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx &=-\frac{\tanh ^{-1}(a x)^3}{3 a^2}+\frac{\int \frac{\tanh ^{-1}(a x)^2}{1-a x} \, dx}{a}\\ &=-\frac{\tanh ^{-1}(a x)^3}{3 a^2}+\frac{\tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^2}-\frac{2 \int \frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=-\frac{\tanh ^{-1}(a x)^3}{3 a^2}+\frac{\tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^2}+\frac{\tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a^2}-\frac{\int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=-\frac{\tanh ^{-1}(a x)^3}{3 a^2}+\frac{\tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{a^2}+\frac{\tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{a^2}-\frac{\text{Li}_3\left (1-\frac{2}{1-a x}\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0647912, size = 68, normalized size = 0.87 \[ -\frac{\tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )-\frac{1}{3} \tanh ^{-1}(a x)^3-\tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{a^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.278, size = 741, normalized size = 9.5 \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{3 \, \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{2} + \log \left (-a x + 1\right )^{3}}{24 \, a^{2}} + \frac{1}{4} \, \int -\frac{a x \log \left (a x + 1\right )^{2} -{\left (3 \, a x + 1\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{3} x^{2} - a}\,{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 \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 \operatorname{atanh}^{2}{\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 \operatorname{artanh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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