Optimal. Leaf size=202 \[ -\frac{8 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{315 a^5}+\frac{a^2 x^7}{252}-\frac{67 x^3}{11340 a^2}+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{29 x}{3780 a^4}+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}-\frac{16 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{315 a^5}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{23 x^5}{3780} \]
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Rubi [A] time = 1.02455, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 59, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {6012, 5916, 5980, 302, 206, 321, 5984, 5918, 2402, 2315} \[ -\frac{8 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{315 a^5}+\frac{a^2 x^7}{252}-\frac{67 x^3}{11340 a^2}+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{29 x}{3780 a^4}+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}-\frac{16 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{315 a^5}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{23 x^5}{3780} \]
Antiderivative was successfully verified.
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Rule 6012
Rule 5916
Rule 5980
Rule 302
Rule 206
Rule 321
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int x^4 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=\int \left (x^4 \tanh ^{-1}(a x)^2-2 a^2 x^6 \tanh ^{-1}(a x)^2+a^4 x^8 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^6 \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^8 \tanh ^{-1}(a x)^2 \, dx+\int x^4 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{1}{5} (2 a) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{7} \left (4 a^3\right ) \int \frac{x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac{1}{9} \left (2 a^5\right ) \int \frac{x^9 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{2 \int x^3 \tanh ^{-1}(a x) \, dx}{5 a}-\frac{2 \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a}-\frac{1}{7} (4 a) \int x^5 \tanh ^{-1}(a x) \, dx+\frac{1}{7} (4 a) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{9} \left (2 a^3\right ) \int x^7 \tanh ^{-1}(a x) \, dx-\frac{1}{9} \left (2 a^3\right ) \int \frac{x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{x^4 \tanh ^{-1}(a x)}{10 a}-\frac{2}{21} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{1}{10} \int \frac{x^4}{1-a^2 x^2} \, dx+\frac{2 \int x \tanh ^{-1}(a x) \, dx}{5 a^3}-\frac{2 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^3}-\frac{4 \int x^3 \tanh ^{-1}(a x) \, dx}{7 a}+\frac{4 \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a}+\frac{1}{9} (2 a) \int x^5 \tanh ^{-1}(a x) \, dx-\frac{1}{9} (2 a) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{21} \left (2 a^2\right ) \int \frac{x^6}{1-a^2 x^2} \, dx-\frac{1}{36} a^4 \int \frac{x^8}{1-a^2 x^2} \, dx\\ &=\frac{x^2 \tanh ^{-1}(a x)}{5 a^3}-\frac{3 x^4 \tanh ^{-1}(a x)}{70 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{\tanh ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{1}{10} \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx+\frac{1}{7} \int \frac{x^4}{1-a^2 x^2} \, dx-\frac{2 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^4}-\frac{4 \int x \tanh ^{-1}(a x) \, dx}{7 a^3}+\frac{4 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a^3}-\frac{\int \frac{x^2}{1-a^2 x^2} \, dx}{5 a^2}+\frac{2 \int x^3 \tanh ^{-1}(a x) \, dx}{9 a}-\frac{2 \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{9 a}-\frac{1}{27} a^2 \int \frac{x^6}{1-a^2 x^2} \, dx+\frac{1}{21} \left (2 a^2\right ) \int \left (-\frac{1}{a^6}-\frac{x^2}{a^4}-\frac{x^4}{a^2}+\frac{1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx-\frac{1}{36} a^4 \int \left (-\frac{1}{a^8}-\frac{x^2}{a^6}-\frac{x^4}{a^4}-\frac{x^6}{a^2}+\frac{1}{a^8 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac{293 x}{1260 a^4}+\frac{41 x^3}{3780 a^2}-\frac{17 x^5}{1260}+\frac{a^2 x^7}{252}-\frac{3 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)-\frac{3 \tanh ^{-1}(a x)^2}{35 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{2 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{5 a^5}-\frac{1}{18} \int \frac{x^4}{1-a^2 x^2} \, dx+\frac{1}{7} \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{36 a^4}+\frac{2 \int \frac{1}{1-a^2 x^2} \, dx}{21 a^4}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{10 a^4}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{5 a^4}+\frac{2 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}+\frac{4 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{7 a^4}+\frac{2 \int x \tanh ^{-1}(a x) \, dx}{9 a^3}-\frac{2 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{9 a^3}+\frac{2 \int \frac{x^2}{1-a^2 x^2} \, dx}{7 a^2}-\frac{1}{27} a^2 \int \left (-\frac{1}{a^6}-\frac{x^2}{a^4}-\frac{x^4}{a^2}+\frac{1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=-\frac{601 x}{3780 a^4}-\frac{277 x^3}{11340 a^2}-\frac{23 x^5}{3780}+\frac{a^2 x^7}{252}-\frac{293 \tanh ^{-1}(a x)}{1260 a^5}+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac{6 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{35 a^5}-\frac{1}{18} \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx-\frac{2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{5 a^5}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{27 a^4}+\frac{\int \frac{1}{1-a^2 x^2} \, dx}{7 a^4}-\frac{2 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{9 a^4}+\frac{2 \int \frac{1}{1-a^2 x^2} \, dx}{7 a^4}-\frac{4 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{7 a^4}-\frac{\int \frac{x^2}{1-a^2 x^2} \, dx}{9 a^2}\\ &=\frac{29 x}{3780 a^4}-\frac{67 x^3}{11340 a^2}-\frac{23 x^5}{3780}+\frac{a^2 x^7}{252}+\frac{601 \tanh ^{-1}(a x)}{3780 a^5}+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{315 a^5}-\frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{5 a^5}+\frac{4 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{7 a^5}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{18 a^4}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{9 a^4}+\frac{2 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{9 a^4}\\ &=\frac{29 x}{3780 a^4}-\frac{67 x^3}{11340 a^2}-\frac{23 x^5}{3780}+\frac{a^2 x^7}{252}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{315 a^5}+\frac{3 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{35 a^5}-\frac{2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{9 a^5}\\ &=\frac{29 x}{3780 a^4}-\frac{67 x^3}{11340 a^2}-\frac{23 x^5}{3780}+\frac{a^2 x^7}{252}-\frac{29 \tanh ^{-1}(a x)}{3780 a^5}+\frac{8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac{4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac{11}{189} a x^6 \tanh ^{-1}(a x)+\frac{1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{315 a^5}-\frac{8 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{315 a^5}\\ \end{align*}
Mathematica [A] time = 1.8055, size = 138, normalized size = 0.68 \[ \frac{288 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+a x \left (45 a^6 x^6-69 a^4 x^4-67 a^2 x^2+87\right )+36 \left (35 a^9 x^9-90 a^7 x^7+63 a^5 x^5-8\right ) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left (105 a^8 x^8-220 a^6 x^6+48 a^4 x^4+96 a^2 x^2-192 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-29\right )}{11340 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.049, size = 259, normalized size = 1.3 \begin{align*}{\frac{{a}^{4}{x}^{9} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{9}}-{\frac{2\,{a}^{2}{x}^{7} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{7}}+{\frac{{x}^{5} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{5}}+{\frac{{a}^{3}{x}^{8}{\it Artanh} \left ( ax \right ) }{36}}-{\frac{11\,a{x}^{6}{\it Artanh} \left ( ax \right ) }{189}}+{\frac{4\,{x}^{4}{\it Artanh} \left ( ax \right ) }{315\,a}}+{\frac{8\,{x}^{2}{\it Artanh} \left ( ax \right ) }{315\,{a}^{3}}}+{\frac{8\,{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{315\,{a}^{5}}}+{\frac{8\,{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{315\,{a}^{5}}}+{\frac{{a}^{2}{x}^{7}}{252}}-{\frac{23\,{x}^{5}}{3780}}-{\frac{67\,{x}^{3}}{11340\,{a}^{2}}}+{\frac{29\,x}{3780\,{a}^{4}}}+{\frac{29\,\ln \left ( ax-1 \right ) }{7560\,{a}^{5}}}-{\frac{29\,\ln \left ( ax+1 \right ) }{7560\,{a}^{5}}}+{\frac{2\, \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{315\,{a}^{5}}}-{\frac{8}{315\,{a}^{5}}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\,\ln \left ( ax-1 \right ) }{315\,{a}^{5}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4}{315\,{a}^{5}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,\ln \left ( ax+1 \right ) }{315\,{a}^{5}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{2\, \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{315\,{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981327, size = 289, normalized size = 1.43 \begin{align*} \frac{1}{22680} \, a^{2}{\left (\frac{90 \, a^{7} x^{7} - 138 \, a^{5} x^{5} - 134 \, a^{3} x^{3} + 174 \, a x - 144 \, \log \left (a x + 1\right )^{2} + 288 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 144 \, \log \left (a x - 1\right )^{2} + 87 \, \log \left (a x - 1\right )}{a^{7}} - \frac{576 \,{\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^{7}} - \frac{87 \, \log \left (a x + 1\right )}{a^{7}}\right )} + \frac{1}{3780} \, a{\left (\frac{105 \, a^{6} x^{8} - 220 \, a^{4} x^{6} + 48 \, a^{2} x^{4} + 96 \, x^{2}}{a^{4}} + \frac{96 \, \log \left (a x + 1\right )}{a^{6}} + \frac{96 \, \log \left (a x - 1\right )}{a^{6}}\right )} \operatorname{artanh}\left (a x\right ) + \frac{1}{315} \,{\left (35 \, a^{4} x^{9} - 90 \, a^{2} x^{7} + 63 \, x^{5}\right )} \operatorname{artanh}\left (a x\right )^{2} \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 ({\left (a^{4} x^{8} - 2 \, a^{2} x^{6} + x^{4}\right )} \operatorname{artanh}\left (a x\right )^{2}, 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 x^{4} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{2}{\left (a x \right )}\, 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{\left (a^{2} x^{2} - 1\right )}^{2} x^{4} \operatorname{artanh}\left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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