Optimal. Leaf size=227 \[ -\frac{3 \text{PolyLog}\left (4,1-\frac{2}{1-a x}\right )}{4 a^4}-\frac{3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 x}{8 a^3 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^4}{4 a^4}-\frac{\tanh ^{-1}(a x)^3}{4 a^4}-\frac{3 \tanh ^{-1}(a x)}{8 a^4}-\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^4} \]
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Rubi [A] time = 0.402791, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6028, 5984, 5918, 5948, 6058, 6062, 6610, 5994, 5956, 199, 206} \[ -\frac{3 \text{PolyLog}\left (4,1-\frac{2}{1-a x}\right )}{4 a^4}-\frac{3 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^4}+\frac{3 \tanh ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{2 a^4}-\frac{3 x}{8 a^3 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^4}{4 a^4}-\frac{\tanh ^{-1}(a x)^3}{4 a^4}-\frac{3 \tanh ^{-1}(a x)}{8 a^4}-\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^4} \]
Antiderivative was successfully verified.
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Rule 6028
Rule 5984
Rule 5918
Rule 5948
Rule 6058
Rule 6062
Rule 6610
Rule 5994
Rule 5956
Rule 199
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3 \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx &=\frac{\int \frac{x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx}{a^2}-\frac{\int \frac{x \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{a^2}\\ &=\frac{\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\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{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^3}\\ &=-\frac{3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^4}+\frac{\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\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 \frac{\tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}+\frac{3 \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}\\ &=\frac{3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^4}+\frac{\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\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{1}{\left (1-a^2 x^2\right )^2} \, dx}{4 a^3}+\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 x}{8 a^3 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^4}+\frac{\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\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{1}{1-a^2 x^2} \, dx}{8 a^3}-\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 x}{8 a^3 \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)}{8 a^4}+\frac{3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac{3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^3}{4 a^4}+\frac{\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\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 \text{Li}_4\left (1-\frac{2}{1-a x}\right )}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.1808, size = 139, normalized size = 0.61 \[ \frac{24 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+24 \tanh ^{-1}(a x) \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+12 \text{PolyLog}\left (4,-e^{-2 \tanh ^{-1}(a x)}\right )-4 \tanh ^{-1}(a x)^4-16 \tanh ^{-1}(a x)^3 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-6 \tanh ^{-1}(a x)^2 \sinh \left (2 \tanh ^{-1}(a x)\right )-3 \sinh \left (2 \tanh ^{-1}(a x)\right )+4 \tanh ^{-1}(a x)^3 \cosh \left (2 \tanh ^{-1}(a x)\right )+6 \tanh ^{-1}(a x) \cosh \left (2 \tanh ^{-1}(a x)\right )}{16 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.348, size = 1015, normalized size = 4.5 \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{{\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )^{4} + 4 \,{\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) - 1\right )} \log \left (-a x + 1\right )^{3}}{64 \,{\left (a^{6} x^{2} - a^{4}\right )}} + \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 x -{\left (3 \, a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \log \left (a x + 1\right ) + 1\right )} \log \left (-a x + 1\right )^{2}}{2 \,{\left (a^{7} x^{4} - 2 \, 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^{4} x^{4} - 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}^{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^{3} \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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