Optimal. Leaf size=119 \[ -\frac{3 x}{8 a \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^2}-\frac{3 \tanh ^{-1}(a x)}{8 a^2} \]
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Rubi [A] time = 0.111713, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5994, 5956, 199, 206} \[ -\frac{3 x}{8 a \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^2}-\frac{3 \tanh ^{-1}(a x)}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 5994
Rule 5956
Rule 199
Rule 206
Rubi steps
\begin{align*} \int \frac{x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx &=\frac{\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{3 \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{2 a}\\ &=-\frac{3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^2}+\frac{\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}+\frac{3}{2} \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^2}+\frac{\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{3 \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx}{4 a}\\ &=-\frac{3 x}{8 a \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^2}+\frac{\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac{3 \int \frac{1}{1-a^2 x^2} \, dx}{8 a}\\ &=-\frac{3 x}{8 a \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)}{8 a^2}+\frac{3 \tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^2}+\frac{\tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0577841, size = 91, normalized size = 0.76 \[ \frac{3 \left (a^2 x^2-1\right ) \log (1-a x)-3 \left (a^2 x^2-1\right ) \log (a x+1)-4 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^3+6 a x+12 a x \tanh ^{-1}(a x)^2-12 \tanh ^{-1}(a x)}{16 a^2 \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.407, size = 1708, normalized size = 14.4 \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 [B] time = 1.00284, size = 402, normalized size = 3.38 \begin{align*} \frac{3 \,{\left (\frac{2 \, x}{a^{2} x^{2} - 1} - \frac{\log \left (a x + 1\right )}{a} + \frac{\log \left (a x - 1\right )}{a}\right )} \operatorname{artanh}\left (a x\right )^{2}}{8 \, a} - \frac{\frac{{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} - 3 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) -{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} - 12 \, a x + 3 \,{\left (2 \, a^{2} x^{2} +{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 2\right )} \log \left (a x + 1\right ) - 6 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a^{2}}{a^{5} x^{2} - a^{3}} - \frac{6 \,{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) +{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 4\right )} a \operatorname{artanh}\left (a x\right )}{a^{4} x^{2} - a^{2}}}{32 \, a} - \frac{\operatorname{artanh}\left (a x\right )^{3}}{2 \,{\left (a^{2} x^{2} - 1\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94574, size = 209, normalized size = 1.76 \begin{align*} \frac{6 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} -{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 12 \, a x - 6 \,{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{32 \,{\left (a^{4} x^{2} - a^{2}\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}^{3}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, 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 )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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