Optimal. Leaf size=64 \[ \frac{1-a^2 x^2}{6 a}+\frac{\log \left (1-a^2 x^2\right )}{3 a}+\frac{1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2}{3} x \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.0212375, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5942, 5910, 260} \[ \frac{1-a^2 x^2}{6 a}+\frac{\log \left (1-a^2 x^2\right )}{3 a}+\frac{1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2}{3} x \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 5942
Rule 5910
Rule 260
Rubi steps
\begin{align*} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx &=\frac{1-a^2 x^2}{6 a}+\frac{1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2}{3} \int \tanh ^{-1}(a x) \, dx\\ &=\frac{1-a^2 x^2}{6 a}+\frac{2}{3} x \tanh ^{-1}(a x)+\frac{1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac{1}{3} (2 a) \int \frac{x}{1-a^2 x^2} \, dx\\ &=\frac{1-a^2 x^2}{6 a}+\frac{2}{3} x \tanh ^{-1}(a x)+\frac{1}{3} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{\log \left (1-a^2 x^2\right )}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0089424, size = 47, normalized size = 0.73 \[ \frac{\log \left (1-a^2 x^2\right )}{3 a}-\frac{1}{3} a^2 x^3 \tanh ^{-1}(a x)-\frac{a x^2}{6}+x \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 48, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}{\it Artanh} \left ( ax \right ){x}^{3}}{3}}+x{\it Artanh} \left ( ax \right ) -{\frac{a{x}^{2}}{6}}+{\frac{\ln \left ( ax-1 \right ) }{3\,a}}+{\frac{\ln \left ( ax+1 \right ) }{3\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954495, size = 63, normalized size = 0.98 \begin{align*} -\frac{1}{6} \,{\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 - \frac{1}{3} \,{\left (a^{2} x^{3} - 3 \, x\right )} \operatorname{artanh}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21157, size = 115, normalized size = 1.8 \begin{align*} -\frac{a^{2} x^{2} +{\left (a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 2 \, \log \left (a^{2} x^{2} - 1\right )}{6 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.995619, size = 49, normalized size = 0.77 \begin{align*} \begin{cases} - \frac{a^{2} x^{3} \operatorname{atanh}{\left (a x \right )}}{3} - \frac{a x^{2}}{6} + x \operatorname{atanh}{\left (a x \right )} + \frac{2 \log{\left (x - \frac{1}{a} \right )}}{3 a} + \frac{2 \operatorname{atanh}{\left (a x \right )}}{3 a} & \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 = 1.16982, size = 69, normalized size = 1.08 \begin{align*} -\frac{1}{6} \, a x^{2} - \frac{1}{6} \,{\left (a^{2} x^{3} - 3 \, x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + \frac{\log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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