Optimal. Leaf size=214 \[ \frac{1}{2} a^4 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )-\frac{1}{2} a^4 \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )-a^4 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )-\frac{a^2}{12 x^2}+\frac{2}{3} a^4 \log \left (1-a^2 x^2\right )+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}-\frac{4}{3} a^4 \log (x)-\frac{3}{4} a^4 \tanh ^{-1}(a x)^2+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac{a \tanh ^{-1}(a x)}{6 x^3}-\frac{\tanh ^{-1}(a x)^2}{4 x^4} \]
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Rubi [A] time = 0.549401, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {6012, 5916, 5982, 266, 44, 36, 29, 31, 5948, 5914, 6052, 6058, 6610} \[ \frac{1}{2} a^4 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )-\frac{1}{2} a^4 \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )-a^4 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )-\frac{a^2}{12 x^2}+\frac{2}{3} a^4 \log \left (1-a^2 x^2\right )+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}-\frac{4}{3} a^4 \log (x)-\frac{3}{4} a^4 \tanh ^{-1}(a x)^2+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac{a \tanh ^{-1}(a x)}{6 x^3}-\frac{\tanh ^{-1}(a x)^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 6012
Rule 5916
Rule 5982
Rule 266
Rule 44
Rule 36
Rule 29
Rule 31
Rule 5948
Rule 5914
Rule 6052
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^5} \, dx &=\int \left (\frac{\tanh ^{-1}(a x)^2}{x^5}-\frac{2 a^2 \tanh ^{-1}(a x)^2}{x^3}+\frac{a^4 \tanh ^{-1}(a x)^2}{x}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x^3} \, dx\right )+a^4 \int \frac{\tanh ^{-1}(a x)^2}{x} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^5} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{1}{2} a \int \frac{\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx-\left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx-\left (4 a^5\right ) \int \frac{\tanh ^{-1}(a x) \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+\frac{1}{2} a \int \frac{\tanh ^{-1}(a x)}{x^4} \, dx+\frac{1}{2} a^3 \int \frac{\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx-\left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^2} \, dx-\left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx-\left (2 a^5\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)}{6 x^3}+\frac{2 a^3 \tanh ^{-1}(a x)}{x}-a^4 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-a^4 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{6} a^2 \int \frac{1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac{1}{2} a^3 \int \frac{\tanh ^{-1}(a x)}{x^2} \, dx-\left (2 a^4\right ) \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx+\frac{1}{2} a^5 \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+a^5 \int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx-a^5 \int \frac{\text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{6 x^3}+\frac{3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac{3}{4} a^4 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-a^4 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a^4 \text{Li}_3\left (1-\frac{2}{1-a x}\right )-\frac{1}{2} a^4 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )+\frac{1}{12} a^2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac{1}{2} a^4 \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx-a^4 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a \tanh ^{-1}(a x)}{6 x^3}+\frac{3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac{3}{4} a^4 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-a^4 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a^4 \text{Li}_3\left (1-\frac{2}{1-a x}\right )-\frac{1}{2} a^4 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )+\frac{1}{12} a^2 \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{a^2}{x}-\frac{a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac{1}{4} a^4 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )-a^4 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-a^6 \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2}{12 x^2}-\frac{a \tanh ^{-1}(a x)}{6 x^3}+\frac{3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac{3}{4} a^4 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{11}{6} a^4 \log (x)+\frac{11}{12} a^4 \log \left (1-a^2 x^2\right )-a^4 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a^4 \text{Li}_3\left (1-\frac{2}{1-a x}\right )-\frac{1}{2} a^4 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )+\frac{1}{4} a^4 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{4} a^6 \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2}{12 x^2}-\frac{a \tanh ^{-1}(a x)}{6 x^3}+\frac{3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac{3}{4} a^4 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{4}{3} a^4 \log (x)+\frac{2}{3} a^4 \log \left (1-a^2 x^2\right )-a^4 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a^4 \text{Li}_3\left (1-\frac{2}{1-a x}\right )-\frac{1}{2} a^4 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )\\ \end{align*}
Mathematica [C] time = 0.359733, size = 238, normalized size = 1.11 \[ \frac{1}{24} \left (24 a^4 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+24 a^4 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+12 a^4 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )-12 a^4 \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-\frac{2 a^2}{x^2}-32 a^4 \log \left (\frac{a x}{\sqrt{1-a^2 x^2}}\right )+\frac{24 a^2 \tanh ^{-1}(a x)^2}{x^2}-16 a^4 \tanh ^{-1}(a x)^3-18 a^4 \tanh ^{-1}(a x)^2+\frac{36 a^3 \tanh ^{-1}(a x)}{x}-24 a^4 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+24 a^4 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+i \pi ^3 a^4+2 a^4-\frac{4 a \tanh ^{-1}(a x)}{x^3}-\frac{6 \tanh ^{-1}(a x)^2}{x^4}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.052, size = 927, normalized size = 4.3 \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 (4 \, a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )^{2}}{16 \, x^{4}} - \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 (4 \, a^{3} x^{3} - a x + 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^{6} - x^{5}\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^{5}}, 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^{5}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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