Optimal. Leaf size=95 \[ \frac{1-a^2 x^2}{12 a^2}+\frac{\log \left (1-a^2 x^2\right )}{6 a^2}-\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}+\frac{x \tanh ^{-1}(a x)}{3 a} \]
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Rubi [A] time = 0.0486081, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5994, 5942, 5910, 260} \[ \frac{1-a^2 x^2}{12 a^2}+\frac{\log \left (1-a^2 x^2\right )}{6 a^2}-\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}+\frac{x \tanh ^{-1}(a x)}{3 a} \]
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 ) \tanh ^{-1}(a x)^2 \, dx &=-\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac{\int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx}{2 a}\\ &=\frac{1-a^2 x^2}{12 a^2}+\frac{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}-\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac{\int \tanh ^{-1}(a x) \, dx}{3 a}\\ &=\frac{1-a^2 x^2}{12 a^2}+\frac{x \tanh ^{-1}(a x)}{3 a}+\frac{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}-\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}-\frac{1}{3} \int \frac{x}{1-a^2 x^2} \, dx\\ &=\frac{1-a^2 x^2}{12 a^2}+\frac{x \tanh ^{-1}(a x)}{3 a}+\frac{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{6 a}-\frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{4 a^2}+\frac{\log \left (1-a^2 x^2\right )}{6 a^2}\\ \end{align*}
Mathematica [A] time = 0.0295572, size = 66, normalized size = 0.69 \[ \frac{-a^2 x^2+2 \log \left (1-a^2 x^2\right )-3 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2+\left (6 a x-2 a^3 x^3\right ) \tanh ^{-1}(a x)}{12 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 185, normalized size = 2. \begin{align*} -{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{4}}{4}}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{2}}{2}}-{\frac{a{\it Artanh} \left ( ax \right ){x}^{3}}{6}}+{\frac{x{\it Artanh} \left ( ax \right ) }{2\,a}}+{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{4\,{a}^{2}}}-{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{4\,{a}^{2}}}+{\frac{ \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{16\,{a}^{2}}}-{\frac{\ln \left ( ax-1 \right ) }{8\,{a}^{2}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{1}{8\,{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 ) }{8\,{a}^{2}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{ \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{16\,{a}^{2}}}-{\frac{{x}^{2}}{12}}+{\frac{\ln \left ( ax-1 \right ) }{6\,{a}^{2}}}+{\frac{\ln \left ( ax+1 \right ) }{6\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965173, size = 100, normalized size = 1.05 \begin{align*} -\frac{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname{artanh}\left (a x\right )^{2}}{4 \, a^{2}} - \frac{{\left (x^{2} - \frac{2 \, \log \left (a x + 1\right )}{a^{2}} - \frac{2 \, \log \left (a x - 1\right )}{a^{2}}\right )} a + 2 \,{\left (a^{2} x^{3} - 3 \, x\right )} \operatorname{artanh}\left (a x\right )}{12 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19842, size = 203, normalized size = 2.14 \begin{align*} -\frac{4 \, a^{2} x^{2} + 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 4 \,{\left (a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 8 \, \log \left (a^{2} x^{2} - 1\right )}{48 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.82278, size = 88, normalized size = 0.93 \begin{align*} \begin{cases} - \frac{a^{2} x^{4} \operatorname{atanh}^{2}{\left (a x \right )}}{4} - \frac{a x^{3} \operatorname{atanh}{\left (a x \right )}}{6} + \frac{x^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{2} - \frac{x^{2}}{12} + \frac{x \operatorname{atanh}{\left (a x \right )}}{2 a} + \frac{\log{\left (x - \frac{1}{a} \right )}}{3 a^{2}} - \frac{\operatorname{atanh}^{2}{\left (a x \right )}}{4 a^{2}} + \frac{\operatorname{atanh}{\left (a x \right )}}{3 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 = 1.16943, size = 115, normalized size = 1.21 \begin{align*} -\frac{1}{16} \,{\left (a^{2} x^{4} - 2 \, x^{2} + \frac{1}{a^{2}}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - \frac{1}{12} \, x^{2} - \frac{1}{12} \,{\left (a x^{3} - \frac{3 \, x}{a}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + \frac{\log \left (a^{2} x^{2} - 1\right )}{6 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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