Optimal. Leaf size=58 \[ \frac{x^2}{2 a^3}+\frac{x}{a^4}+\frac{1}{2 a^5 (1-a x)}+\frac{7 \log (1-a x)}{4 a^5}+\frac{\log (a x+1)}{4 a^5} \]
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Rubi [A] time = 0.115073, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 88} \[ \frac{x^2}{2 a^3}+\frac{x}{a^4}+\frac{1}{2 a^5 (1-a x)}+\frac{7 \log (1-a x)}{4 a^5}+\frac{\log (a x+1)}{4 a^5} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac{x^4}{(1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac{1}{a^4}+\frac{x}{a^3}+\frac{1}{2 a^4 (-1+a x)^2}+\frac{7}{4 a^4 (-1+a x)}+\frac{1}{4 a^4 (1+a x)}\right ) \, dx\\ &=\frac{x}{a^4}+\frac{x^2}{2 a^3}+\frac{1}{2 a^5 (1-a x)}+\frac{7 \log (1-a x)}{4 a^5}+\frac{\log (1+a x)}{4 a^5}\\ \end{align*}
Mathematica [A] time = 0.060876, size = 45, normalized size = 0.78 \[ \frac{2 \left (a^2 x^2+2 a x+\frac{1}{1-a x}\right )+7 \log (1-a x)+\log (a x+1)}{4 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 49, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2\,{a}^{3}}}+{\frac{x}{{a}^{4}}}+{\frac{\ln \left ( ax+1 \right ) }{4\,{a}^{5}}}-{\frac{1}{2\,{a}^{5} \left ( ax-1 \right ) }}+{\frac{7\,\ln \left ( ax-1 \right ) }{4\,{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94287, size = 70, normalized size = 1.21 \begin{align*} -\frac{1}{2 \,{\left (a^{6} x - a^{5}\right )}} + \frac{a x^{2} + 2 \, x}{2 \, a^{4}} + \frac{\log \left (a x + 1\right )}{4 \, a^{5}} + \frac{7 \, \log \left (a x - 1\right )}{4 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77871, size = 144, normalized size = 2.48 \begin{align*} \frac{2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 4 \, a x +{\left (a x - 1\right )} \log \left (a x + 1\right ) + 7 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \,{\left (a^{6} x - a^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.426931, size = 48, normalized size = 0.83 \begin{align*} - \frac{1}{2 a^{6} x - 2 a^{5}} + \frac{x^{2}}{2 a^{3}} + \frac{x}{a^{4}} + \frac{\frac{7 \log{\left (x - \frac{1}{a} \right )}}{4} + \frac{\log{\left (x + \frac{1}{a} \right )}}{4}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16013, size = 76, normalized size = 1.31 \begin{align*} \frac{\log \left ({\left | a x + 1 \right |}\right )}{4 \, a^{5}} + \frac{7 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a^{5}} + \frac{a^{3} x^{2} + 2 \, a^{2} x}{2 \, a^{6}} - \frac{1}{2 \,{\left (a x - 1\right )} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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