Optimal. Leaf size=90 \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{1}{4} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+3 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]
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Rubi [A] time = 0.272736, antiderivative size = 90, 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 = {5982, 5916, 5988, 5932, 5948, 6056, 6610} \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{1}{4} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+3 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 5982
Rule 5916
Rule 5988
Rule 5932
Rule 5948
Rule 6056
Rule 6610
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx+\int \frac{\tanh ^{-1}(a x)^3}{x^2} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{1}{4} a \tanh ^{-1}(a x)^4+(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx\\ &=a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{1}{4} a \tanh ^{-1}(a x)^4+(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx\\ &=a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{1}{4} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-\left (6 a^2\right ) \int \frac{\tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{1}{4} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\left (3 a^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{1}{4} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{3}{2} a \text{Li}_3\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}
Mathematica [C] time = 0.193665, size = 93, normalized size = 1.03 \[ -a \left (-3 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-\frac{1}{4} \tanh ^{-1}(a x)^4+\frac{\tanh ^{-1}(a x)^3}{a x}+\tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\frac{i \pi ^3}{8}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.343, size = 826, normalized size = 9.2 \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{a x \log \left (-a x + 1\right )^{4} - 4 \,{\left (a x \log \left (a x + 1\right ) + 2 \, a x - 2\right )} \log \left (-a x + 1\right )^{3} + 6 \,{\left (a x \log \left (a x + 1\right )^{2} - 4 \,{\left (a x + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{64 \, x} - \frac{1}{8} \, \int \frac{2 \, \log \left (a x + 1\right )^{3} + 3 \,{\left ({\left (a^{3} x^{3} + a^{2} x^{2} - 2\right )} \log \left (a x + 1\right )^{2} - 4 \,{\left (a^{3} x^{3} + 2 \, a^{2} x^{2} + a x\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{2 \,{\left (a^{2} x^{4} - x^{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{\operatorname{artanh}\left (a x\right )^{3}}{a^{2} x^{4} - x^{2}}, 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{\operatorname{atanh}^{3}{\left (a x \right )}}{a^{2} x^{4} - x^{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{\operatorname{artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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