Optimal. Leaf size=170 \[ \frac{5 a^6}{504 x^2}+\frac{a^4}{84 x^4}-\frac{a^2}{168 x^6}-\frac{2}{63} a^8 \log \left (1-a^2 x^2\right )-\frac{a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}+\frac{4}{63} a^8 \log (x)+\frac{1}{24} a^8 \tanh ^{-1}(a x)^2-\frac{a^7 \tanh ^{-1}(a x)}{12 x}-\frac{a \tanh ^{-1}(a x)}{28 x^7}-\frac{\tanh ^{-1}(a x)^2}{8 x^8} \]
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Rubi [A] time = 0.840865, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 56, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6012, 5916, 5982, 266, 44, 36, 29, 31, 5948} \[ \frac{5 a^6}{504 x^2}+\frac{a^4}{84 x^4}-\frac{a^2}{168 x^6}-\frac{2}{63} a^8 \log \left (1-a^2 x^2\right )-\frac{a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}+\frac{4}{63} a^8 \log (x)+\frac{1}{24} a^8 \tanh ^{-1}(a x)^2-\frac{a^7 \tanh ^{-1}(a x)}{12 x}-\frac{a \tanh ^{-1}(a x)}{28 x^7}-\frac{\tanh ^{-1}(a x)^2}{8 x^8} \]
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
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^9} \, dx &=\int \left (\frac{\tanh ^{-1}(a x)^2}{x^9}-\frac{2 a^2 \tanh ^{-1}(a x)^2}{x^7}+\frac{a^4 \tanh ^{-1}(a x)^2}{x^5}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x^7} \, dx\right )+a^4 \int \frac{\tanh ^{-1}(a x)^2}{x^5} \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^9} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{1}{4} a \int \frac{\tanh ^{-1}(a x)}{x^8 \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^6 \left (1-a^2 x^2\right )} \, dx+\frac{1}{2} a^5 \int \frac{\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{1}{4} a \int \frac{\tanh ^{-1}(a x)}{x^8} \, dx+\frac{1}{4} a^3 \int \frac{\tanh ^{-1}(a x)}{x^6 \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^6} \, dx+\frac{1}{2} a^5 \int \frac{\tanh ^{-1}(a x)}{x^4} \, dx-\frac{1}{3} \left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac{1}{2} a^7 \int \frac{\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{28 x^7}+\frac{2 a^3 \tanh ^{-1}(a x)}{15 x^5}-\frac{a^5 \tanh ^{-1}(a x)}{6 x^3}-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{1}{28} a^2 \int \frac{1}{x^7 \left (1-a^2 x^2\right )} \, dx+\frac{1}{4} a^3 \int \frac{\tanh ^{-1}(a x)}{x^6} \, dx-\frac{1}{15} \left (2 a^4\right ) \int \frac{1}{x^5 \left (1-a^2 x^2\right )} \, dx+\frac{1}{4} a^5 \int \frac{\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x^4} \, dx+\frac{1}{6} a^6 \int \frac{1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac{1}{2} a^7 \int \frac{\tanh ^{-1}(a x)}{x^2} \, dx-\frac{1}{3} \left (2 a^7\right ) \int \frac{\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac{1}{2} a^9 \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{28 x^7}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}+\frac{a^5 \tanh ^{-1}(a x)}{18 x^3}-\frac{a^7 \tanh ^{-1}(a x)}{2 x}+\frac{1}{4} a^8 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{1}{56} a^2 \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac{1}{20} a^4 \int \frac{1}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac{1}{15} a^4 \operatorname{Subst}\left (\int \frac{1}{x^3 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac{1}{4} a^5 \int \frac{\tanh ^{-1}(a x)}{x^4} \, dx+\frac{1}{12} a^6 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{9} \left (2 a^6\right ) \int \frac{1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac{1}{4} a^7 \int \frac{\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 a^7\right ) \int \frac{\tanh ^{-1}(a x)}{x^2} \, dx+\frac{1}{2} a^8 \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 a^9\right ) \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{28 x^7}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac{a^5 \tanh ^{-1}(a x)}{36 x^3}+\frac{a^7 \tanh ^{-1}(a x)}{6 x}-\frac{1}{12} a^8 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{1}{56} a^2 \operatorname{Subst}\left (\int \left (\frac{1}{x^4}+\frac{a^2}{x^3}+\frac{a^4}{x^2}+\frac{a^6}{x}-\frac{a^8}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac{1}{40} a^4 \operatorname{Subst}\left (\int \frac{1}{x^3 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{15} a^4 \operatorname{Subst}\left (\int \left (\frac{1}{x^3}+\frac{a^2}{x^2}+\frac{a^4}{x}-\frac{a^6}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac{1}{12} a^6 \int \frac{1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac{1}{12} a^6 \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}{9} a^6 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac{1}{4} a^7 \int \frac{\tanh ^{-1}(a x)}{x^2} \, dx+\frac{1}{4} a^8 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{3} \left (2 a^8\right ) \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx+\frac{1}{4} a^9 \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac{a^2}{168 x^6}+\frac{41 a^4}{1680 x^4}-\frac{29 a^6}{840 x^2}-\frac{a \tanh ^{-1}(a x)}{28 x^7}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac{a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac{a^7 \tanh ^{-1}(a x)}{12 x}+\frac{1}{24} a^8 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{29}{420} a^8 \log (x)-\frac{29}{840} a^8 \log \left (1-a^2 x^2\right )+\frac{1}{40} a^4 \operatorname{Subst}\left (\int \left (\frac{1}{x^3}+\frac{a^2}{x^2}+\frac{a^4}{x}-\frac{a^6}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac{1}{24} a^6 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{9} a^6 \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^8 \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx+\frac{1}{4} a^8 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{3} a^8 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac{1}{4} a^{10} \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2}{168 x^6}+\frac{a^4}{84 x^4}+\frac{13 a^6}{252 x^2}-\frac{a \tanh ^{-1}(a x)}{28 x^7}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac{a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac{a^7 \tanh ^{-1}(a x)}{12 x}+\frac{1}{24} a^8 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{25}{63} a^8 \log (x)-\frac{25}{126} a^8 \log \left (1-a^2 x^2\right )+\frac{1}{24} a^6 \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}{8} a^8 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{3} a^8 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{3} a^{10} \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2}{168 x^6}+\frac{a^4}{84 x^4}+\frac{5 a^6}{504 x^2}-\frac{a \tanh ^{-1}(a x)}{28 x^7}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac{a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac{a^7 \tanh ^{-1}(a x)}{12 x}+\frac{1}{24} a^8 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}-\frac{47}{252} a^8 \log (x)+\frac{47}{504} a^8 \log \left (1-a^2 x^2\right )+\frac{1}{8} a^8 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{8} a^{10} \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2}{168 x^6}+\frac{a^4}{84 x^4}+\frac{5 a^6}{504 x^2}-\frac{a \tanh ^{-1}(a x)}{28 x^7}+\frac{a^3 \tanh ^{-1}(a x)}{12 x^5}-\frac{a^5 \tanh ^{-1}(a x)}{36 x^3}-\frac{a^7 \tanh ^{-1}(a x)}{12 x}+\frac{1}{24} a^8 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{8 x^8}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^6}-\frac{a^4 \tanh ^{-1}(a x)^2}{4 x^4}+\frac{4}{63} a^8 \log (x)-\frac{2}{63} a^8 \log \left (1-a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.064794, size = 124, normalized size = 0.73 \[ \frac{a^2 x^2 \left (5 a^4 x^4+6 a^2 x^2+32 a^6 x^6 \log (x)-16 a^6 x^6 \log \left (1-a^2 x^2\right )-3\right )+21 \left (a^2 x^2+3\right ) \left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)^2-2 a x \left (21 a^6 x^6+7 a^4 x^4-21 a^2 x^2+9\right ) \tanh ^{-1}(a x)}{504 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 253, normalized size = 1.5 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{8\,{x}^{8}}}-{\frac{{a}^{4} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{4\,{x}^{4}}}+{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3\,{x}^{6}}}-{\frac{{a}^{8}{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{24}}-{\frac{a{\it Artanh} \left ( ax \right ) }{28\,{x}^{7}}}+{\frac{{a}^{3}{\it Artanh} \left ( ax \right ) }{12\,{x}^{5}}}-{\frac{{a}^{5}{\it Artanh} \left ( ax \right ) }{36\,{x}^{3}}}-{\frac{{a}^{7}{\it Artanh} \left ( ax \right ) }{12\,x}}+{\frac{{a}^{8}{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{24}}-{\frac{{a}^{8} \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{96}}+{\frac{{a}^{8}\ln \left ( ax-1 \right ) }{48}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{{a}^{8} \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{96}}+{\frac{{a}^{8}\ln \left ( ax+1 \right ) }{48}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{{a}^{8}}{48}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{2\,{a}^{8}\ln \left ( ax-1 \right ) }{63}}-{\frac{{a}^{2}}{168\,{x}^{6}}}+{\frac{{a}^{4}}{84\,{x}^{4}}}+{\frac{5\,{a}^{6}}{504\,{x}^{2}}}+{\frac{4\,{a}^{8}\ln \left ( ax \right ) }{63}}-{\frac{2\,{a}^{8}\ln \left ( ax+1 \right ) }{63}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974479, size = 275, normalized size = 1.62 \begin{align*} \frac{1}{2016} \,{\left (128 \, a^{6} \log \left (x\right ) - \frac{21 \, a^{6} x^{6} \log \left (a x + 1\right )^{2} + 21 \, a^{6} x^{6} \log \left (a x - 1\right )^{2} + 64 \, a^{6} x^{6} \log \left (a x - 1\right ) - 20 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 2 \,{\left (21 \, a^{6} x^{6} \log \left (a x - 1\right ) - 32 \, a^{6} x^{6}\right )} \log \left (a x + 1\right ) + 12}{x^{6}}\right )} a^{2} + \frac{1}{504} \,{\left (21 \, a^{7} \log \left (a x + 1\right ) - 21 \, a^{7} \log \left (a x - 1\right ) - \frac{2 \,{\left (21 \, a^{6} x^{6} + 7 \, a^{4} x^{4} - 21 \, a^{2} x^{2} + 9\right )}}{x^{7}}\right )} a \operatorname{artanh}\left (a x\right ) - \frac{{\left (6 \, a^{4} x^{4} - 8 \, a^{2} x^{2} + 3\right )} \operatorname{artanh}\left (a x\right )^{2}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99544, size = 338, normalized size = 1.99 \begin{align*} -\frac{64 \, a^{8} x^{8} \log \left (a^{2} x^{2} - 1\right ) - 128 \, a^{8} x^{8} \log \left (x\right ) - 20 \, a^{6} x^{6} - 24 \, a^{4} x^{4} + 12 \, a^{2} x^{2} - 21 \,{\left (a^{8} x^{8} - 6 \, a^{4} x^{4} + 8 \, a^{2} x^{2} - 3\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 4 \,{\left (21 \, a^{7} x^{7} + 7 \, a^{5} x^{5} - 21 \, a^{3} x^{3} + 9 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{2016 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.916, size = 168, normalized size = 0.99 \begin{align*} \begin{cases} \frac{4 a^{8} \log{\left (x \right )}}{63} - \frac{4 a^{8} \log{\left (x - \frac{1}{a} \right )}}{63} + \frac{a^{8} \operatorname{atanh}^{2}{\left (a x \right )}}{24} - \frac{4 a^{8} \operatorname{atanh}{\left (a x \right )}}{63} - \frac{a^{7} \operatorname{atanh}{\left (a x \right )}}{12 x} + \frac{5 a^{6}}{504 x^{2}} - \frac{a^{5} \operatorname{atanh}{\left (a x \right )}}{36 x^{3}} - \frac{a^{4} \operatorname{atanh}^{2}{\left (a x \right )}}{4 x^{4}} + \frac{a^{4}}{84 x^{4}} + \frac{a^{3} \operatorname{atanh}{\left (a x \right )}}{12 x^{5}} + \frac{a^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{3 x^{6}} - \frac{a^{2}}{168 x^{6}} - \frac{a \operatorname{atanh}{\left (a x \right )}}{28 x^{7}} - \frac{\operatorname{atanh}^{2}{\left (a x \right )}}{8 x^{8}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \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^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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