Optimal. Leaf size=157 \[ -\frac{8}{15} a^5 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{a^2}{30 x^3}+\frac{7 a^3 \tanh ^{-1}(a x)}{15 x^2}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{11 a^4}{30 x}+\frac{8}{15} a^5 \tanh ^{-1}(a x)^2-\frac{11}{30} a^5 \tanh ^{-1}(a x)-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+\frac{16}{15} a^5 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{10 x^4}-\frac{\tanh ^{-1}(a x)^2}{5 x^5} \]
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Rubi [A] time = 0.594464, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6012, 5916, 5982, 325, 206, 5988, 5932, 2447} \[ -\frac{8}{15} a^5 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{a^2}{30 x^3}+\frac{7 a^3 \tanh ^{-1}(a x)}{15 x^2}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{11 a^4}{30 x}+\frac{8}{15} a^5 \tanh ^{-1}(a x)^2-\frac{11}{30} a^5 \tanh ^{-1}(a x)-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+\frac{16}{15} a^5 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{10 x^4}-\frac{\tanh ^{-1}(a x)^2}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 6012
Rule 5916
Rule 5982
Rule 325
Rule 206
Rule 5988
Rule 5932
Rule 2447
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^6} \, dx &=\int \left (\frac{\tanh ^{-1}(a x)^2}{x^6}-\frac{2 a^2 \tanh ^{-1}(a x)^2}{x^4}+\frac{a^4 \tanh ^{-1}(a x)^2}{x^2}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x^4} \, dx\right )+a^4 \int \frac{\tanh ^{-1}(a x)^2}{x^2} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^6} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+\frac{1}{5} (2 a) \int \frac{\tanh ^{-1}(a x)}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (4 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx+\left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+\frac{1}{5} (2 a) \int \frac{\tanh ^{-1}(a x)}{x^5} \, dx+\frac{1}{5} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (4 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^3} \, dx-\frac{1}{3} \left (4 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{10 x^4}+\frac{2 a^3 \tanh ^{-1}(a x)}{3 x^2}+\frac{1}{3} a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+2 a^5 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+\frac{1}{10} a^2 \int \frac{1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac{1}{5} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^3} \, dx-\frac{1}{3} \left (2 a^4\right ) \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac{1}{5} \left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (4 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\left (2 a^6\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2}{30 x^3}+\frac{2 a^4}{3 x}-\frac{a \tanh ^{-1}(a x)}{10 x^4}+\frac{7 a^3 \tanh ^{-1}(a x)}{15 x^2}+\frac{8}{15} a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+\frac{2}{3} a^5 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )-a^5 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\frac{1}{10} a^4 \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac{1}{5} a^4 \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac{1}{5} \left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac{1}{3} \left (2 a^6\right ) \int \frac{1}{1-a^2 x^2} \, dx+\frac{1}{3} \left (4 a^6\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2}{30 x^3}+\frac{11 a^4}{30 x}-\frac{2}{3} a^5 \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{10 x^4}+\frac{7 a^3 \tanh ^{-1}(a x)}{15 x^2}+\frac{8}{15} a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+\frac{16}{15} a^5 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )-\frac{1}{3} a^5 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\frac{1}{10} a^6 \int \frac{1}{1-a^2 x^2} \, dx+\frac{1}{5} a^6 \int \frac{1}{1-a^2 x^2} \, dx-\frac{1}{5} \left (2 a^6\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2}{30 x^3}+\frac{11 a^4}{30 x}-\frac{11}{30} a^5 \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{10 x^4}+\frac{7 a^3 \tanh ^{-1}(a x)}{15 x^2}+\frac{8}{15} a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{x}+\frac{16}{15} a^5 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )-\frac{8}{15} a^5 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}
Mathematica [A] time = 0.761504, size = 118, normalized size = 0.75 \[ \frac{-16 a^5 x^5 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )+a^2 x^2 \left (11 a^2 x^2-1\right )+2 (a x-1)^3 \left (8 a^2 x^2+9 a x+3\right ) \tanh ^{-1}(a x)^2+a x \tanh ^{-1}(a x) \left (-11 a^4 x^4+14 a^2 x^2+32 a^4 x^4 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-3\right )}{30 x^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.066, size = 272, normalized size = 1.7 \begin{align*} -{\frac{{a}^{4} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{5\,{x}^{5}}}+{\frac{2\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{8\,{a}^{5}{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{15}}-{\frac{a{\it Artanh} \left ( ax \right ) }{10\,{x}^{4}}}+{\frac{7\,{a}^{3}{\it Artanh} \left ( ax \right ) }{15\,{x}^{2}}}+{\frac{16\,{a}^{5}{\it Artanh} \left ( ax \right ) \ln \left ( ax \right ) }{15}}-{\frac{8\,{a}^{5}{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{15}}+{\frac{11\,{a}^{5}\ln \left ( ax-1 \right ) }{60}}-{\frac{{a}^{2}}{30\,{x}^{3}}}+{\frac{11\,{a}^{4}}{30\,x}}-{\frac{11\,{a}^{5}\ln \left ( ax+1 \right ) }{60}}-{\frac{8\,{a}^{5}{\it dilog} \left ( ax \right ) }{15}}-{\frac{8\,{a}^{5}{\it dilog} \left ( ax+1 \right ) }{15}}-{\frac{8\,{a}^{5}\ln \left ( ax \right ) \ln \left ( ax+1 \right ) }{15}}-{\frac{2\,{a}^{5} \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{15}}+{\frac{8\,{a}^{5}}{15}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,{a}^{5}\ln \left ( ax-1 \right ) }{15}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{2\,{a}^{5} \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{15}}+{\frac{4\,{a}^{5}}{15}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\,{a}^{5}\ln \left ( ax+1 \right ) }{15}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988413, size = 323, normalized size = 2.06 \begin{align*} \frac{1}{60} \,{\left (32 \,{\left (\log \left (a x - 1\right ) \log \left (\frac{1}{2} \, a x + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, a x + \frac{1}{2}\right )\right )} a^{3} - 32 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )} a^{3} + 32 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )} a^{3} - 11 \, a^{3} \log \left (a x + 1\right ) + 11 \, a^{3} \log \left (a x - 1\right ) + \frac{2 \,{\left (4 \, a^{3} x^{3} \log \left (a x + 1\right )^{2} - 8 \, a^{3} x^{3} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 4 \, a^{3} x^{3} \log \left (a x - 1\right )^{2} + 11 \, a^{2} x^{2} - 1\right )}}{x^{3}}\right )} a^{2} - \frac{1}{30} \,{\left (16 \, a^{4} \log \left (a^{2} x^{2} - 1\right ) - 16 \, a^{4} \log \left (x^{2}\right ) - \frac{14 \, a^{2} x^{2} - 3}{x^{4}}\right )} a \operatorname{artanh}\left (a x\right ) - \frac{{\left (15 \, a^{4} x^{4} - 10 \, a^{2} x^{2} + 3\right )} \operatorname{artanh}\left (a x\right )^{2}}{15 \, x^{5}} \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^{6}}, 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^{6}}\, 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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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