Optimal. Leaf size=138 \[ \frac{\left (1-a^2 x^2\right )^2}{60 a^2}+\frac{2 \left (1-a^2 x^2\right )}{45 a^2}+\frac{4 \log \left (1-a^2 x^2\right )}{45 a^2}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}+\frac{4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac{8 x \tanh ^{-1}(a x)}{45 a} \]
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Rubi [A] time = 0.088121, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5994, 5942, 5910, 260} \[ \frac{\left (1-a^2 x^2\right )^2}{60 a^2}+\frac{2 \left (1-a^2 x^2\right )}{45 a^2}+\frac{4 \log \left (1-a^2 x^2\right )}{45 a^2}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}+\frac{4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac{8 x \tanh ^{-1}(a x)}{45 a} \]
Antiderivative was successfully verified.
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Rule 5994
Rule 5942
Rule 5910
Rule 260
Rubi steps
\begin{align*} \int x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac{\int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx}{3 a}\\ &=\frac{\left (1-a^2 x^2\right )^2}{60 a^2}+\frac{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac{4 \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx}{15 a}\\ &=\frac{2 \left (1-a^2 x^2\right )}{45 a^2}+\frac{\left (1-a^2 x^2\right )^2}{60 a^2}+\frac{4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac{8 \int \tanh ^{-1}(a x) \, dx}{45 a}\\ &=\frac{2 \left (1-a^2 x^2\right )}{45 a^2}+\frac{\left (1-a^2 x^2\right )^2}{60 a^2}+\frac{8 x \tanh ^{-1}(a x)}{45 a}+\frac{4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}-\frac{8}{45} \int \frac{x}{1-a^2 x^2} \, dx\\ &=\frac{2 \left (1-a^2 x^2\right )}{45 a^2}+\frac{\left (1-a^2 x^2\right )^2}{60 a^2}+\frac{8 x \tanh ^{-1}(a x)}{45 a}+\frac{4 x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{45 a}+\frac{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{15 a}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{6 a^2}+\frac{4 \log \left (1-a^2 x^2\right )}{45 a^2}\\ \end{align*}
Mathematica [A] time = 0.053743, size = 82, normalized size = 0.59 \[ \frac{3 a^4 x^4-14 a^2 x^2+16 \log \left (1-a^2 x^2\right )+4 a x \left (3 a^4 x^4-10 a^2 x^2+15\right ) \tanh ^{-1}(a x)+30 \left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)^2}{180 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 219, normalized size = 1.6 \begin{align*}{\frac{{a}^{4} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{6}}{6}}-{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{4}}{2}}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{2}}{2}}+{\frac{{a}^{3}{\it Artanh} \left ( ax \right ){x}^{5}}{15}}-{\frac{2\,a{\it Artanh} \left ( ax \right ){x}^{3}}{9}}+{\frac{x{\it Artanh} \left ( ax \right ) }{3\,a}}+{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{6\,{a}^{2}}}-{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{6\,{a}^{2}}}+{\frac{ \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{24\,{a}^{2}}}-{\frac{\ln \left ( ax-1 \right ) }{12\,{a}^{2}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{1}{12\,{a}^{2}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{\ln \left ( ax+1 \right ) }{12\,{a}^{2}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{ \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{24\,{a}^{2}}}+{\frac{{a}^{2}{x}^{4}}{60}}-{\frac{7\,{x}^{2}}{90}}+{\frac{4\,\ln \left ( ax-1 \right ) }{45\,{a}^{2}}}+{\frac{4\,\ln \left ( ax+1 \right ) }{45\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964607, size = 126, normalized size = 0.91 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname{artanh}\left (a x\right )^{2}}{6 \, a^{2}} + \frac{{\left (3 \, a^{2} x^{4} - 14 \, x^{2} + \frac{16 \, \log \left (a x + 1\right )}{a^{2}} + \frac{16 \, \log \left (a x - 1\right )}{a^{2}}\right )} a + 4 \,{\left (3 \, a^{4} x^{5} - 10 \, a^{2} x^{3} + 15 \, x\right )} \operatorname{artanh}\left (a x\right )}{180 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00645, size = 261, normalized size = 1.89 \begin{align*} \frac{6 \, a^{4} x^{4} - 28 \, a^{2} x^{2} + 15 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 4 \,{\left (3 \, a^{5} x^{5} - 10 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + 32 \, \log \left (a^{2} x^{2} - 1\right )}{360 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.57061, size = 133, normalized size = 0.96 \begin{align*} \begin{cases} \frac{a^{4} x^{6} \operatorname{atanh}^{2}{\left (a x \right )}}{6} + \frac{a^{3} x^{5} \operatorname{atanh}{\left (a x \right )}}{15} - \frac{a^{2} x^{4} \operatorname{atanh}^{2}{\left (a x \right )}}{2} + \frac{a^{2} x^{4}}{60} - \frac{2 a x^{3} \operatorname{atanh}{\left (a x \right )}}{9} + \frac{x^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{2} - \frac{7 x^{2}}{90} + \frac{x \operatorname{atanh}{\left (a x \right )}}{3 a} + \frac{8 \log{\left (x - \frac{1}{a} \right )}}{45 a^{2}} - \frac{\operatorname{atanh}^{2}{\left (a x \right )}}{6 a^{2}} + \frac{8 \operatorname{atanh}{\left (a x \right )}}{45 a^{2}} & \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 [A] time = 1.20939, size = 151, normalized size = 1.09 \begin{align*} \frac{1}{60} \, a^{2} x^{4} + \frac{1}{24} \,{\left (a^{4} x^{6} - 3 \, a^{2} x^{4} + 3 \, x^{2} - \frac{1}{a^{2}}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - \frac{7}{90} \, x^{2} + \frac{1}{90} \,{\left (3 \, a^{3} x^{5} - 10 \, a x^{3} + \frac{15 \, x}{a}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + \frac{4 \, \log \left (a^{2} x^{2} - 1\right )}{45 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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