Optimal. Leaf size=186 \[ \frac{1}{2} \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )-\frac{1}{2} \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )-\tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )+\tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )+\frac{a^2 x^2}{12}-\frac{2}{3} \log \left (1-a^2 x^2\right )+\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+\frac{1}{6} a^3 x^3 \tanh ^{-1}(a x)-a^2 x^2 \tanh ^{-1}(a x)^2-\frac{3}{2} a x \tanh ^{-1}(a x)+\frac{3}{4} \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right ) \]
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Rubi [A] time = 0.52983, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {6012, 5914, 6052, 5948, 6058, 6610, 5916, 5980, 5910, 260, 266, 43} \[ \frac{1}{2} \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )-\frac{1}{2} \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )-\tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )+\tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )+\frac{a^2 x^2}{12}-\frac{2}{3} \log \left (1-a^2 x^2\right )+\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+\frac{1}{6} a^3 x^3 \tanh ^{-1}(a x)-a^2 x^2 \tanh ^{-1}(a x)^2-\frac{3}{2} a x \tanh ^{-1}(a x)+\frac{3}{4} \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right ) \]
Antiderivative was successfully verified.
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Rule 6012
Rule 5914
Rule 6052
Rule 5948
Rule 6058
Rule 6610
Rule 5916
Rule 5980
Rule 5910
Rule 260
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x} \, dx &=\int \left (\frac{\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+a^4 x^3 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^3 \tanh ^{-1}(a x)^2 \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x} \, dx\\ &=-a^2 x^2 \tanh ^{-1}(a x)^2+\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-(4 a) \int \frac{\tanh ^{-1}(a x) \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (2 a^3\right ) \int \frac{x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac{1}{2} a^5 \int \frac{x^4 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-a^2 x^2 \tanh ^{-1}(a x)^2+\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-(2 a) \int \tanh ^{-1}(a x) \, dx+(2 a) \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+(2 a) \int \frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx-(2 a) \int \frac{\tanh ^{-1}(a x) \log \left (2-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx+\frac{1}{2} a^3 \int x^2 \tanh ^{-1}(a x) \, dx-\frac{1}{2} a^3 \int \frac{x^2 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-2 a x \tanh ^{-1}(a x)+\frac{1}{6} a^3 x^3 \tanh ^{-1}(a x)+\tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a \int \tanh ^{-1}(a x) \, dx-\frac{1}{2} a \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+a \int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx-a \int \frac{\text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (2 a^2\right ) \int \frac{x}{1-a^2 x^2} \, dx-\frac{1}{6} a^4 \int \frac{x^3}{1-a^2 x^2} \, dx\\ &=-\frac{3}{2} a x \tanh ^{-1}(a x)+\frac{1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac{3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} \text{Li}_3\left (1-\frac{2}{1-a x}\right )-\frac{1}{2} \text{Li}_3\left (-1+\frac{2}{1-a x}\right )-\frac{1}{2} a^2 \int \frac{x}{1-a^2 x^2} \, dx-\frac{1}{12} a^4 \operatorname{Subst}\left (\int \frac{x}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{3}{2} a x \tanh ^{-1}(a x)+\frac{1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac{3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{3}{4} \log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} \text{Li}_3\left (1-\frac{2}{1-a x}\right )-\frac{1}{2} \text{Li}_3\left (-1+\frac{2}{1-a x}\right )-\frac{1}{12} a^4 \operatorname{Subst}\left (\int \left (-\frac{1}{a^2}-\frac{1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 x^2}{12}-\frac{3}{2} a x \tanh ^{-1}(a x)+\frac{1}{6} a^3 x^3 \tanh ^{-1}(a x)+\frac{3}{4} \tanh ^{-1}(a x)^2-a^2 x^2 \tanh ^{-1}(a x)^2+\frac{1}{4} a^4 x^4 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{2}{3} \log \left (1-a^2 x^2\right )-\tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} \text{Li}_3\left (1-\frac{2}{1-a x}\right )-\frac{1}{2} \text{Li}_3\left (-1+\frac{2}{1-a x}\right )\\ \end{align*}
Mathematica [A] time = 0.0632203, size = 191, normalized size = 1.03 \[ -\frac{1}{2} \text{PolyLog}\left (3,\frac{-a x-1}{a x-1}\right )+\frac{1}{2} \text{PolyLog}\left (3,\frac{a x+1}{a x-1}\right )+\tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{-a x-1}{a x-1}\right )-\tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{a x+1}{a x-1}\right )+\frac{a^2 x^2}{12}-\frac{2}{3} \log \left (1-a^2 x^2\right )+\frac{1}{6} a x \left (a^2 x^2+3\right ) \tanh ^{-1}(a x)-\left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2+\frac{1}{4} \left (a^4 x^4-1\right ) \tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 1.165, size = 728, normalized size = 3.9 \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 (a^{4} x^{4} - 4 \, a^{2} x^{2}\right )} \log \left (-a x + 1\right )^{2} - \frac{1}{4} \, \int -\frac{2 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right )^{2} -{\left (a^{5} x^{5} - 4 \, a^{3} x^{3} + 4 \,{\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{2 \,{\left (a x^{2} - x\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{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{2}}{x}, 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}\, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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