Optimal. Leaf size=162 \[ -\frac{2 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{35 a^5}-\frac{2 x^3}{315 a^2}-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac{4 x}{105 a^4}+\frac{2 \tanh ^{-1}(a x)^2}{35 a^5}-\frac{4 \tanh ^{-1}(a x)}{105 a^5}-\frac{4 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a^5}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2+\frac{x^4 \tanh ^{-1}(a x)}{35 a}-\frac{x^5}{105} \]
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Rubi [A] time = 0.579493, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6014, 5916, 5980, 302, 206, 321, 5984, 5918, 2402, 2315} \[ -\frac{2 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{35 a^5}-\frac{2 x^3}{315 a^2}-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac{4 x}{105 a^4}+\frac{2 \tanh ^{-1}(a x)^2}{35 a^5}-\frac{4 \tanh ^{-1}(a x)}{105 a^5}-\frac{4 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a^5}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2+\frac{x^4 \tanh ^{-1}(a x)}{35 a}-\frac{x^5}{105} \]
Antiderivative was successfully verified.
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Rule 6014
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 ) \tanh ^{-1}(a x)^2 \, dx &=-\left (a^2 \int x^6 \tanh ^{-1}(a x)^2 \, dx\right )+\int x^4 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{1}{7} a^2 x^7 \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 (2 a^3\right ) \int \frac{x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{1}{7} a^2 x^7 \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} (2 a) \int x^5 \tanh ^{-1}(a x) \, dx+\frac{1}{7} (2 a) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{x^4 \tanh ^{-1}(a x)}{10 a}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{1}{7} a^2 x^7 \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{2 \int x^3 \tanh ^{-1}(a x) \, dx}{7 a}+\frac{2 \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a}+\frac{1}{21} a^2 \int \frac{x^6}{1-a^2 x^2} \, dx\\ &=\frac{x^2 \tanh ^{-1}(a x)}{5 a^3}+\frac{x^4 \tanh ^{-1}(a x)}{35 a}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{\tanh ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac{1}{14} \int \frac{x^4}{1-a^2 x^2} \, dx-\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{2 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^4}-\frac{2 \int x \tanh ^{-1}(a x) \, dx}{7 a^3}+\frac{2 \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{1}{21} 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{53 x}{210 a^4}+\frac{11 x^3}{630 a^2}-\frac{x^5}{105}+\frac{2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac{x^4 \tanh ^{-1}(a x)}{35 a}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac{2 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{5 a^5}+\frac{1}{14} \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}{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{\tanh ^{-1}(a x)}{1-a x} \, dx}{7 a^4}+\frac{2 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}+\frac{\int \frac{x^2}{1-a^2 x^2} \, dx}{7 a^2}\\ &=\frac{4 x}{105 a^4}-\frac{2 x^3}{315 a^2}-\frac{x^5}{105}-\frac{53 \tanh ^{-1}(a x)}{210 a^5}+\frac{2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac{x^4 \tanh ^{-1}(a x)}{35 a}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac{4 \tanh ^{-1}(a x) \log \left (\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 )}{5 a^5}+\frac{\int \frac{1}{1-a^2 x^2} \, dx}{14 a^4}+\frac{\int \frac{1}{1-a^2 x^2} \, dx}{7 a^4}-\frac{2 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{7 a^4}\\ &=\frac{4 x}{105 a^4}-\frac{2 x^3}{315 a^2}-\frac{x^5}{105}-\frac{4 \tanh ^{-1}(a x)}{105 a^5}+\frac{2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac{x^4 \tanh ^{-1}(a x)}{35 a}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac{4 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{35 a^5}-\frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{5 a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{7 a^5}\\ &=\frac{4 x}{105 a^4}-\frac{2 x^3}{315 a^2}-\frac{x^5}{105}-\frac{4 \tanh ^{-1}(a x)}{105 a^5}+\frac{2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac{x^4 \tanh ^{-1}(a x)}{35 a}-\frac{1}{21} a x^6 \tanh ^{-1}(a x)+\frac{2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac{1}{5} x^5 \tanh ^{-1}(a x)^2-\frac{1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac{4 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{35 a^5}-\frac{2 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{35 a^5}\\ \end{align*}
Mathematica [A] time = 0.902051, size = 113, normalized size = 0.7 \[ -\frac{-18 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+3 a^5 x^5+2 a^3 x^3+9 \left (5 a^7 x^7-7 a^5 x^5+2\right ) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left (5 a^6 x^6-3 a^4 x^4-6 a^2 x^2+12 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+4\right )-12 a x}{315 a^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.049, size = 225, normalized size = 1.4 \begin{align*} -{\frac{{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{x}^{6}{\it Artanh} \left ( ax \right ) }{21}}+{\frac{{x}^{4}{\it Artanh} \left ( ax \right ) }{35\,a}}+{\frac{2\,{x}^{2}{\it Artanh} \left ( ax \right ) }{35\,{a}^{3}}}+{\frac{2\,{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{35\,{a}^{5}}}+{\frac{2\,{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{35\,{a}^{5}}}+{\frac{ \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{70\,{a}^{5}}}-{\frac{2}{35\,{a}^{5}}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{\ln \left ( ax-1 \right ) }{35\,{a}^{5}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{1}{35\,{a}^{5}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{\ln \left ( ax+1 \right ) }{35\,{a}^{5}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{ \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{70\,{a}^{5}}}-{\frac{{x}^{5}}{105}}-{\frac{2\,{x}^{3}}{315\,{a}^{2}}}+{\frac{4\,x}{105\,{a}^{4}}}+{\frac{2\,\ln \left ( ax-1 \right ) }{105\,{a}^{5}}}-{\frac{2\,\ln \left ( ax+1 \right ) }{105\,{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983634, size = 257, normalized size = 1.59 \begin{align*} -\frac{1}{630} \, a^{2}{\left (\frac{6 \, a^{5} x^{5} + 4 \, a^{3} x^{3} - 24 \, a x + 9 \, \log \left (a x + 1\right )^{2} - 18 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 9 \, \log \left (a x - 1\right )^{2} - 12 \, \log \left (a x - 1\right )}{a^{7}} + \frac{36 \,{\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{12 \, \log \left (a x + 1\right )}{a^{7}}\right )} - \frac{1}{105} \, a{\left (\frac{5 \, a^{4} x^{6} - 3 \, a^{2} x^{4} - 6 \, x^{2}}{a^{4}} - \frac{6 \, \log \left (a x + 1\right )}{a^{6}} - \frac{6 \, \log \left (a x - 1\right )}{a^{6}}\right )} \operatorname{artanh}\left (a x\right ) - \frac{1}{35} \,{\left (5 \, a^{2} x^{7} - 7 \, 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^{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} \operatorname{atanh}^{2}{\left (a x \right )}\, dx - \int a^{2} x^{6} \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 )} 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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