Optimal. Leaf size=162 \[ -a^2 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )+a^2 \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )+2 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )-2 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )+\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2+a^2 \log (x)+a^3 x \tanh ^{-1}(a x)-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-\frac{a \tanh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.461116, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {6012, 5916, 5982, 266, 36, 29, 31, 5948, 5914, 6052, 6058, 6610, 5980, 5910, 260} \[ -a^2 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )+a^2 \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )+2 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )-2 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )+\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2+a^2 \log (x)+a^3 x \tanh ^{-1}(a x)-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-\frac{a \tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 6012
Rule 5916
Rule 5982
Rule 266
Rule 36
Rule 29
Rule 31
Rule 5948
Rule 5914
Rule 6052
Rule 6058
Rule 6610
Rule 5980
Rule 5910
Rule 260
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^3} \, dx &=\int \left (\frac{\tanh ^{-1}(a x)^2}{x^3}-\frac{2 a^2 \tanh ^{-1}(a x)^2}{x}+a^4 x \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x} \, dx\right )+a^4 \int x \tanh ^{-1}(a x)^2 \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^3} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a \int \frac{\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+\left (8 a^3\right ) \int \frac{\tanh ^{-1}(a x) \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx-a^5 \int \frac{x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a \int \frac{\tanh ^{-1}(a x)}{x^2} \, dx+a^3 \int \tanh ^{-1}(a x) \, dx-\left (4 a^3\right ) \int \frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (4 a^3\right ) \int \frac{\tanh ^{-1}(a x) \log \left (2-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{x}+a^3 x \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+2 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )-2 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+a^2 \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx-\left (2 a^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (2 a^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx-a^4 \int \frac{x}{1-a^2 x^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{x}+a^3 x \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+2 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )-2 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )-a^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )+a^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a \tanh ^{-1}(a x)}{x}+a^3 x \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+2 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )-2 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )-a^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )+a^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} a^4 \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a \tanh ^{-1}(a x)}{x}+a^3 x \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^4 x^2 \tanh ^{-1}(a x)^2-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a^2 \log (x)+2 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )-2 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )-a^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )+a^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )\\ \end{align*}
Mathematica [A] time = 0.0699802, size = 183, normalized size = 1.13 \[ a^2 \text{PolyLog}\left (3,\frac{-a x-1}{a x-1}\right )-a^2 \text{PolyLog}\left (3,\frac{a x+1}{a x-1}\right )-2 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{-a x-1}{a x-1}\right )+2 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{a x+1}{a x-1}\right )+\frac{1}{2} a^2 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2+\frac{\left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)+a^3 x \tanh ^{-1}(a x)-4 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{a \tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.15, size = 774, normalized size = 4.8 \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{1}{16} \,{\left (2 \, x^{2} \log \left (-a x + 1\right ) - a{\left (\frac{a x^{2} + 2 \, x}{a^{2}} + \frac{2 \, \log \left (a x - 1\right )}{a^{3}}\right )}\right )} a^{4} - \frac{1}{2} \, a^{4} \int x \log \left (a x + 1\right ) \log \left (-a x + 1\right )\,{d x} + \frac{1}{4} \, a^{3} \int a x \log \left (a x + 1\right )^{2}\,{d x} + \frac{1}{4} \, a^{3} \int \frac{\log \left (a x + 1\right )^{2}}{a^{3} x^{3}}\,{d x} + \frac{1}{4} \,{\left (a x -{\left (a x - 1\right )} \log \left (-a x + 1\right ) - 1\right )} a^{2} - \frac{1}{2} \, a^{2} \int \frac{\log \left (a x + 1\right )^{2}}{x}\,{d x} + a^{2} \int \frac{\log \left (a x + 1\right ) \log \left (-a x + 1\right )}{x}\,{d x} - \frac{1}{4} \, a^{2} \int \frac{\log \left (-a x + 1\right )}{x}\,{d x} - \frac{1}{4} \,{\left (a{\left (\log \left (a x - 1\right ) - \log \left (x\right )\right )} - \frac{\log \left (-a x + 1\right )}{x}\right )} a + \frac{{\left (a^{4} x^{4} - 1\right )} \log \left (-a x + 1\right )^{2}}{8 \, x^{2}} - \frac{1}{2} \, \int \frac{\log \left (a x + 1\right ) \log \left (-a x + 1\right )}{x^{3}}\,{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{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{2}}{x^{3}}, 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{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{x^{3}}\, 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{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname{artanh}\left (a x\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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