Optimal. Leaf size=104 \[ \frac{\left (1-a^2 x^2\right )^2}{20 a}+\frac{2 \left (1-a^2 x^2\right )}{15 a}+\frac{4 \log \left (1-a^2 x^2\right )}{15 a}+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{8}{15} x \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.0437777, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5942, 5910, 260} \[ \frac{\left (1-a^2 x^2\right )^2}{20 a}+\frac{2 \left (1-a^2 x^2\right )}{15 a}+\frac{4 \log \left (1-a^2 x^2\right )}{15 a}+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{8}{15} x \tanh ^{-1}(a x) \]
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 )^2 \tanh ^{-1}(a x) \, dx &=\frac{\left (1-a^2 x^2\right )^2}{20 a}+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{4}{5} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx\\ &=\frac{2 \left (1-a^2 x^2\right )}{15 a}+\frac{\left (1-a^2 x^2\right )^2}{20 a}+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{8}{15} \int \tanh ^{-1}(a x) \, dx\\ &=\frac{2 \left (1-a^2 x^2\right )}{15 a}+\frac{\left (1-a^2 x^2\right )^2}{20 a}+\frac{8}{15} x \tanh ^{-1}(a x)+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)-\frac{1}{15} (8 a) \int \frac{x}{1-a^2 x^2} \, dx\\ &=\frac{2 \left (1-a^2 x^2\right )}{15 a}+\frac{\left (1-a^2 x^2\right )^2}{20 a}+\frac{8}{15} x \tanh ^{-1}(a x)+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{4 \log \left (1-a^2 x^2\right )}{15 a}\\ \end{align*}
Mathematica [A] time = 0.0166325, size = 71, normalized size = 0.68 \[ \frac{a^3 x^4}{20}+\frac{4 \log \left (1-a^2 x^2\right )}{15 a}+\frac{1}{5} a^4 x^5 \tanh ^{-1}(a x)-\frac{2}{3} a^2 x^3 \tanh ^{-1}(a x)-\frac{7 a x^2}{30}+x \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 68, normalized size = 0.7 \begin{align*}{\frac{{a}^{4}{\it Artanh} \left ( ax \right ){x}^{5}}{5}}-{\frac{2\,{a}^{2}{\it Artanh} \left ( ax \right ){x}^{3}}{3}}+x{\it Artanh} \left ( ax \right ) +{\frac{{x}^{4}{a}^{3}}{20}}-{\frac{7\,a{x}^{2}}{30}}+{\frac{4\,\ln \left ( ax-1 \right ) }{15\,a}}+{\frac{4\,\ln \left ( ax+1 \right ) }{15\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975285, size = 89, normalized size = 0.86 \begin{align*} \frac{1}{60} \,{\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 + \frac{1}{15} \,{\left (3 \, a^{4} x^{5} - 10 \, a^{2} x^{3} + 15 \, 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 = 1.95209, size = 161, normalized size = 1.55 \begin{align*} \frac{3 \, a^{4} x^{4} - 14 \, a^{2} x^{2} + 2 \,{\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 ) + 16 \, \log \left (a^{2} x^{2} - 1\right )}{60 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.77116, size = 75, normalized size = 0.72 \begin{align*} \begin{cases} \frac{a^{4} x^{5} \operatorname{atanh}{\left (a x \right )}}{5} + \frac{a^{3} x^{4}}{20} - \frac{2 a^{2} x^{3} \operatorname{atanh}{\left (a x \right )}}{3} - \frac{7 a x^{2}}{30} + x \operatorname{atanh}{\left (a x \right )} + \frac{8 \log{\left (x - \frac{1}{a} \right )}}{15 a} + \frac{8 \operatorname{atanh}{\left (a x \right )}}{15 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 = 1.16773, size = 103, normalized size = 0.99 \begin{align*} \frac{1}{30} \,{\left (3 \, a^{4} x^{5} - 10 \, a^{2} x^{3} + 15 \, x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + \frac{4 \, \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{15 \, a} + \frac{3 \, a^{7} x^{4} - 14 \, a^{5} x^{2}}{60 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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